According to this article in Philosophy Now, Sartre wrote this as a piece of polemic and apologia for athiestic humanism; which is possibly one reason why he later regretted its publication.
Polemic doesn't tend to argue discursively, backing up each claim credibly; it tends to go for greater rhetorical effect - and this means argumentation and reasoning is dropped; it also plays to Satres strength as a specifically literary kind of philosopher - how many philosophers have written plays and novels?
So, in one sense it's not surprising that you've exposed a hole in Sartres thinking.
In this essay, Sartre is defending his conception of existentialism against a number of charges brought against it:
too focused on the individual, without thinking on the relation between the individual and society, and of human solidarity - a charge made alike by Christians and Marxists.
too pessimistic and dystopic; focusing on the ignominious condition of man; in Being & Nothingness he calls man a 'useless passion' and that all forms of sexual love were doomed to masochism or sadism - Hegels master-slave dialectic in the context of the personal
Sartre takes freedom - the individual free will - as an 'absolute', meaning he begins with that; what 'we choose is always for the better' says as you say, that the individual choosing his actions is evidence that he is choosing for the best.
His next move to the universal, is in a sense a lyrical and oracular move, he veils himself in the voice of all, and speaks for all - note the use of 'we'; so this is not man, the individual whose fate he's concerned with here, and nor men in their plurality and variety, but man as a species, and therefore as an essence (these are terms in Aristotles categories, that are used in a different way in Aristotelianism; and a term that he reverses in his abandonment of God: existence before essence, not after; for Sartre,
a man defines his essence by his acts acting in freedom).
This is where his literariness, his poetics come in; it works only if one accepts this as a kind of assertion of destiny; as a kind of prophesying.
Discursively, though it's questionable exactly on the basis that you've made; and it's the same critique the article levels at him.
His claim doesn't stand up within the text; but this ignores the nature of the essay as a polemic and defence; the question is whether he provides a defence of this position elsewhere in his works.
The article suggests not, and that he's unable to demonstrate that his ethics, in the terms he's framed them, doesn't lead to moral anarchy.
A closer reading, however may show that he's taken on something of the legislative burden of the Kantian moral code - to only act, as though you're acting for all; and then it would be a question how he's used the scaffolding from Kant to buttress his own position - I'm not enough of an expert to answer this; but hopefully this might prompt someone to offer a better answer.