That's perfectly fine in English

Added now.

No, not really. 'szczęście' is 'luck' indeed, but 'szczęśliwy' is in fact 'happy'.

It may also be something that brings luck (Scrooge McDuck's "szczęśliwa dziesięciocentówka", the Lucky Dime).

But "to be lucky" is in Polish "mieć szczęście" (lit. to have luck).

wesoły means happy in the sense of being "cheerful." szczęśliwy means happy as in a person who is "happy in life"

This is not correct on many levels, in my opinion – first, English isn't 'more flexible' than Polish, yes there are some things that are way easier to say in English than Polish, but in my experience, they are more than evened out by things that are way easier to state in Polish than English.
Second, there aren't 'many more' idioms in English than in Polish – fact is, most European(at least those on the Catholic side of Catholic-Orthodox schism) languages that were 'standardised' before Gutenberg invented printing press, have about the same amount of idiomatic phrases(and also a similar word count of about one million, but that's beside the point), whether it's French, German, Polish or English(and so on), simply because they have a similarly sized 'backlog' of literary works since the Middle Ages.

And this is very good example of the first, actually: „On nigdy nie był szczęśliwy, ale mimo to zawsze był wesoły”, yields a bit idiotic "He was never happy but despite that, he was always happy", unless you will start using less common words like 'content', 'joyful', 'cheerful' but all these words obviously also have their more specific and less common analogues in Polish… :P Despite the fact both translate to happy, „wesoły” ≠ „szczęśliwy”, as va-diim already pointed out to you.

I agree about the English. One can also be a wesoły człowiek in Polish, a chronically cheerful person, but szczęśliwy człowiek doesn't necessarily mean he is all smiles, fun, and rainbows. It means that he has good luck in life and is generally emotionally happy, not just in a happy mood. Wesoły​ tends to be an acute condition, and szczęśliwy tends to be a chronic one.