„A putea a bea cafea” does not work. Difficult to say why, but it's probably because in a normal sentence where you use the indicative form of the verb „a putea” it is almost always followed by the conjunctive form. Examples (with other verbs as well):
(Eu) pot să te ajut dacă vrei. - I can help you if you want.
N-ai voie să te ridici! - You are not allowed to stand up!
Trebuie să-ți faci treaba! - You must/you have to do your job!
Am nevoie să știu/Trebuie să știu! (same meaning) - I must know!
M-am dus să vorbesc cu profesorul. - I went to speak with the teacher/professor.
These verbs and others are almost always followed by a verb in the conjunctive. The infinitive form could also be used (only for some of these verbs!), but it usually sounds strange/poetic. I would avoid it.
Yes, the infinitive isn't used all that much nowadays. Depending on the context, „să pot să beau cafea” seems like the best choice.
Mi-aș dori să pot să beau cafea. - I would like to be able to drink coffee (a better translation: I wish I were able to drink coffee).
Even inversions are possible:
Să pot să mănânc carne ar fi minunat! Din păcate nu am voie. - Being able to eat meat would be wonderful! I am not allowed to, though. (because you are suffering from a disease, for instance).
As a rule of thumb: If you are unsure between the infinitive and the conjunctive, just use the conjunctive. Chances are you are right.
As a Romanian native speaker, I hear the short infinitive form almost daily. I don't think it is that weird, especially when a general statement is done: "A putea ajuta pe cineva" vs "Eu pot să te ajut"
As a simple rule, "a putea" can be followed by either the conjuctive form or by the short infinitive form (without "a").
This is absolutely correct, but I just feel that teaching people to use the short infinitive in these situations would not be right. You are walking on thin ice, since using the short infinitive form works some of the time, whereas using the subjunctive (conjunctive) works (almost) all the time.
In your examples, both "a putea ajuta pe cineva" and "a putea să ajuți pe cineva" work, but in the second case, only "eu pot să te ajut" works. "A putea a te ajuta" is definitely not correct.
It is quite a complicated discussion overall and picking the right form requires a certain amount of finesse, however using the subjunctive is usually the safer bet here.