More "simple consideration" needed
Empirically, there seems to be no true happiness without some sort of suffering
Define "true happiness" and "suffering". Until you can express what you mean by those terms, you aren't ready to ask the question - and you certainly shouldn't ask other people to answer the question.
Your examples don't cut it as explanation because they're factually and logically wrong.
happiness given by chemical drugs is not long lasting and always has a destructive effect
Not necessarily. Sure, drugs only last for as long as they're in your system, but there's nothing impossible about keeping them topped up. There's no absolute reason for all drugs to have some negative side-effects, only body physiology, and body physiology is not a moral framework.
happiness and fulfilment often come from overcoming something
And why does that count as "suffering"? I feel happy and fulfilled when I've built something around my house. I've overcome that challenge. But I don't have to dislike the building process in order to feel happy and fulfilled.
For a really obvious example of happiness being unrelated to any of that, consider love. Simply being with the other person is all that's necessary for happiness. Doesn't need drugs, isn't related to obstacles before they met or obstacles during the relationship, can be long lasting, isn't destructive. (Love isn't destructive, only other feelings such as jealousy which are ascribed to "love" but really aren't.) Hell, it doesn't even have to be a person - you can love your pets, and you can even love your concept of a god.
More thought needed about who says it's a "happy life"
it is easy to think of people who have had an overall unhappy life (severe physical conditions, discriminations, life events...)
By your terms, looking from the outside, you might think so. Do those factors make them think they have an overall unhappy life, though? Some do, some don't. And conversely, some people may have extreme privilege (sports stars, movie stars, pop stars) and still consider themselves to have an unhappy life, even though they don't have any of the issues that your so-called "unhappy" people do.
Do you think you have any right, responsibility or ability to pass judgement on whether someone's life is "overall happy"?
More research needed
I guess no one ever argued that one can be "simply happy"
Of course they have. The entire religion/philosophy of Buddhism is centred around it. The key concept is that this world and its obstacles is fundamentally incapable of satisfying you, and true happiness comes from spiritual revelation arising from religious/mystical practises. Jesus said something similar with: "Whoever comes to me will never be hungry again. Whoever believes in me will never be thirsty."
Or in a more secular way, consider the poem Desiderata. The point of that poem isn't telling you to overcome obstacles, it's to inspire you to remain happy regardless of obstacles.