Some possible cultural roots of U.S. happiness and Western Continental Europe suffering below.
We could easily find counter-examples to the generalizations outlined in this answer: Voltaire was a playful and optimist hedonist, the liberal Austrian school of economics was... Austrian (but they were obliged to migrate because of nazism), Belgium has a whole tradition of joyful and humorous comicbooks, etc.
This answer is based on the postulate that everything is a question of proportion, given that hardly anything is black or white.
Overall, the trends outlined in this answer are deemed to weigh stronger than the counter-examples/cultural counter-balances that could be opposed to them. For instance, it could be arguable that in France the influence of Rousseau is stronger than the influence of Voltaire, although both were very famous, influential and productive.
And the same is true for other regions. Both daoism and confucianism were born in China, but today, in the Popular Republic of China, the influence, of confucianism is stronger than the influence of daoism, if the latter has still any influence at all.
One manifestation of that is when are compared American Romanticism (= Transcendentalism) and European Romanticism. Psychological suffering is not a theme in Transcendentalism, but it is a major one in European Romanticism (German one, French one, and also English one). The European romantics somehow romanticized negative emotional states.
"Melancholy is a disease which consists in seeing things as they are." (Gérard de Nerval, "Paradoxe et vérité", 1844) Fr. "La mélancolie est une maladie qui consiste à voir les choses comme elles sont."
A quick search for "most famous european painting" on Google gives, for modern pieces, Picasso's "Guernica", Géricault's "The Raft of the Medusa", Munch's "The Scream".
A quick search for "most famous american painting" on Google gives, for modern pieces, Hopper's "Nighthawks" and Leutze's "Washington Crossing the Delaware".
- The figures of the (tortured) artist and the poète maudit
The figure of the artist, sacralized by the European romantics, may be stronger in Europe than in the U.S., where the figure of the entrepreneur, is maybe stronger.
Since for European romantics, the artist was a tortured one, and that art/knowledge/truth is all a matter of beauty (i.e. appearance) (cf. SEP), then we could suppute that (psychological) suffering tends to be cool in Europe.
The tradition of the cool tortured but popular and successful artist, started with Charles Baudelaire, was pursued by Serge Gainsbourg, and now Michel Houellebecq.
To that must be added the figure of the tortured unpopular artist, but no less genius, with Van Gogh, Modigliani, Satie, Nietzsche, etc.
A particular experimental artistic work on the relation between madness (and marginality in general) and art was conducted by the "outsider artist" Jean Dubuffet.
In relation to the "tortured artist" figure is the poète maudit (lit. "cursed poet") one, developed by French poets Alfred de Vigny in his 1832 novel Stello, and Paul Verlaine in his essay Les Poètes maudits in 1884. Other poètes maudits include Charles Baudelaire, Arthur Rimbaud, Lautréamont, Alice de Chambrier (all French), and the American Hart Crane.
"A poète maudit is a poet living a life outside or against society. Abuse of drugs and alcohol, insanity, crime, violence, and in general any societal sin, often resulting in an early death, are typical elements of the biography of a poète maudit." (Wikipedia)
Another manifestation of that is the themes of the Continental philosophy tradition (e.g. Nietzsche, Camus, Schopenhauer, Foucault) which is very concentrated on power struggle, absurdity of the human condition, and suffering, while the U.S. philosophical tradition is marked by pragmatism and the analytic philosophy which focus on how to solve concrete problems.
The recognition of the negative aspects of human condition has been pointed in the commentary as a mark of realism on the part of these philosophers, in contrast to the U.S. happiness pursuit deemed to be irrealist. Ironically, these same philosophers (and artists, see above) are often anti-realist when it comes to the capacity of science to apprehend the world and provide solutions and answers to the human condition issues. Scientific progress (from human sciences to natural sciences and engineering) has been driven mainly in the U.S., at least for the last decades.
Wonder whether the comparatively greater influence of socialism (Babeuf and Marx) which sees life in society as an affair of constant collective power struggle of the masses undergoing unfair treatment plays a role (in comparison, the individualist classical liberalism of Locke and Thomas Jefferson has a greater place in the U.S. compared to the Western Continental Europe's socialism tradition).
Continental political philosophy is marked by a strong pessimism.
Many philosophers (Plessner, Arendt, Adorno, Meillassoux, and others) more or less related to the Continental philosophy tradition are "pretty explicit that the culture of uncritical optimism which started with the transition from Enlightenment to Romanticism had historically been one of the root causes of totalitarian tendencies" [credits to Philip Klöcking, quoted, for providing these information in the comments].
A strong pessimism about the possibilities offered by economic liberalism also exists in the Continental economic philosophy.
Karl Polanyi believed economic liberalism drives towards fascist barbarism.
Marx, who himself recognized and praised the immense accomplishment of capitalism, advocated for its abolition.
While there are arguments to say that the first ecologist was Rousseau, we find the same line of idea in some contemporary ecologist writings: "There is no doubt that the world has experienced a transformation in material wellbeing in the past two hundred years, and Pinker documents this in detail, from the increased availability of clothing, food, and transportation, to the seemingly mundane yet enormously important decrease in the cost of artificial light. However, there is a point where the rise in economic activity begins to decouple from wellbeing." Jeremy Lent.
And today's ecologists are no less pessimistic.
Western Continental Europe's is historically Catholicism which is heavily centered around suffering (cf. Jesus's statement "If any man would come after me... let him take up his cross daily"). I don't know the role suffering has in English and American Protestantism though.
It might be an example of interaction between the different realms, in that catholicism and communism may be intertwined spiritually and historically.
Wonder to what extent the culture of historical European peoples such as the Vikings (Scandinavia), or the Gauls (France), whose reputation is to be "bloodthirsty raiders navigating the northern seas" (Vikings) and "brave barbarian, living in a primitive state", embodying "the good savage" (Gauls), still have an influence in the identity imaginary of actual Europeans. In France, the Gauls cultural heritage is transmitted to younger generations via the comic book (and its cinematographic adaptations) "Asterix and Obelix" (to give an idea of its impact in the French society, it may be like Superman/Spiderman/Batman in the U.S.). The prowesses of this modern Viking are promoted on a large scale.
Not mentioning the imaginary around Middle Ages history in Europe (cf. the knights and the Crusades).
The American cowboys seem cute in comparison.
As to how it transcribes concretely in the contemporean European cultures, and how it would relate to the happiness/suffering culture, I am not sure.
In Europe, rugby is a popular sport that could be compared to American football, but in rugby players don't wear any protection. The French Légion Étrangère (independent from the regular French army) receives a lot of attention from the general public, while their training mindset, and therefore all the imaginary surrounding it, is very Spartan. Seems in the U.S. military, a great focus is given to technological and technical prowess. For instance, Chris Kyle, reknown and highly respected for his extraordinary technical skill as a sniper.
More sophisticated and delightful sports and cultures were historically reserved to the elite (classical music, fine arts, fencing, gastronomy) in Europe.
- Treatment of psychological suffering
In France (TTBOMK it is not the case in Germany or Belgium for instance), psychoanalysis still enjoys a strong recognition and diffusion (including by cultural elites) among the society. However, psychoanalysis has been found to be a pseudo-science, or at the very least, its model has been almost completely abandoned in the field of experimental psychology and psychiatry in the U.S. (but also all over the world, including Europe and France), where every theory about the human psychology is based on the scientific empirical method, and where the treatment of mental illness is most often the cognitive behavioral therapy, or other treatments taking into account multiple factors such as biology, personal habits, social ties, etc. In contrast, in psychonalysis, depression in children is for instance understood within the myth of Oedipus.
Maybe symptomatically, as of April 2023, the Wikipedia page for the "Dual process theory", which has now largely supplanted the psychoanalytic concept of "subconscious", has no translation in any European language (but is available in English, and Middle-East and East-Asian languages).
Therefore, many cases of mental illness might be left untreated.
Disclaimer: That doesn't mean Western Continental Europe is not influenced by the U.S. way of thinking described (and it is greatly influenced by the U.S.), changing somehow the habits and states of mind (through American pop culture, e.g. Disneyland Paris, American comedies).
Neither that the U.S. is not influenced by the Western Continental Europe way ot thinking described (and it is greatly, increasingly?, influenced by it), changing somehow the habits and states of mind, themselves reinforcing back Western Continental Europe (examples of that is the very popular Nietzschean movie "Fight Club" (1999), or the movie "Whiplash" (2014), directed by the French-American Damien Chazelle. As of a more general influence of European culture on American culture and going further back in time is the influence of European Symbolism, Dadaism and Surrealism on Bob Dylan).