That's a tricky sentence. "Faz outra vez" is actually how most people say it, but it's grammatically incorrect. We shouldn't use the present tense but the imperative "Faça outra vez".
So what happens is we usually write it in the imperative, but say it in the present tense because the imperative sounds too aggressive.
Yeah, you're right about that. The imperative for tu is faz, although we actually mean it as the present tense for você, so at least in places where people don't use tu, that's how we interpret it. And if you're using it in writing, it only works if the rest of your text uses tu.
Actually, we use "vai" in a sense of "go somewhere", so I don't think it would be very appropriate. "Faz de novo" is something we say more often, but - a little tip - we usually ask here, like "faz de novo?", because it sounds a bit aggressive to "command" someone to do it again. But it depends on the context.
Faz outra vez - Do it again (?)
"Do it another time" refers to the future as defined in American English:
A: Shall we meet for lunch. B: Sorry, I'm busy. Let's do it another time.
"Make more time" - because "to make time" means to organize your time so there's more time to do something else. In Portuguese that would be something like "organize o seu tempo", or "faça o seu tempo render". "Faze-lo outra vez" - if you mean "fazê-lo", that would literally "to do it again". In my opinion, the best translation for "faz outra vez!" this would be "do it again!"
It's not really out of politeness, it's just that in many regions of Brazil the imperative is not "properly" used. Speakers prefer to use "faz" instead of "faça", for some reason I don't really know how to explain. However, in the northeast you will most definitely hear "faça" instead of "faz". Here's an interesting link about it: https://ciberduvidas.iscte-iul.pt/consultorio/perguntas/faz-e-faze/32768
In fact, there are "two forms" of imperative.
One related to "tu", which is "faz".
Another one related to "você", which is "faça".
None of them is wrong, both are indeed imperative forms.
In Rio, "faz de novo" is indeed something we would use more for friends and family, while "faça de novo" is something your teacher or boss would tell you.