I think this is stretching it, but the Coleman map is the collections of the fn, so in a sense the analog of the Coleman map in the geometric case is the isomorphism f. What is the correct way to look at the Coleman map is an excellent question, which admits a precise albeit technical question: the Coleman map is an instance of the so-called epsilon morphism. You could read for instance Fukaya-Kato on this. It is natural to feel intimated by the level of Fukaya-Kato or Kato's lecture, but you should give it a try once in a while: you will learn a lot from them.

But $A$ is not local, or am I missing something?

For instance, Boyarsky suggestion, though interesting, is a nice example of an insight that seems to me to have completely permeated the mathematical community: I am pretty sure the idea that induction could be used effectively on primes was known to me as an undergraduate.

Your rules 1 and 3 set the bar almost impossibly high: valuable mathematical insights contained in a paper 110 years old at least have in all likelihood permeated the mathematical community by now, so that requiring that you would never have acquired them by reading more recent sources seems almost contradictory. Anyway, I have always had an even broader understanding of the saying than A.Putman: I understand it as "read the original research material, not the derivative works". And I think this is a great advice!

FWIW, R.Diestel writes in Graph Theory that the original approach of Appel-Haken "has not been immune to criticism, not only because of their use of a computer".

Brian may have high standards of what is interesting and almost certainly has very high standards of what is obvious, but I, at least, have been grateful that Mazur and Rubin made this remark. After all, it means that Kolyvagin systems are useful to study the ETNC for any principal artinian ring, and this is not uninteresting if one is interested in interactions between the ETNC and deformation theory.

Sure, but the question was about p-adic L-functions in general, not specifically for elliptic curves.

That said, if your question has some definite purpose, perhaps it could help if you explained how such a modular forms would (or would not) help. Presumably, if you have a construction that produces a certain output specifically at primes, this construction tells you something about a_p, and this is exactly what I lack in my (very amateur) understanding of the problem.

A common heuristic is to regard the coefficients a_n as random with respect to congruence. So a_p already has a very slight chance of being divisible by p, let alone zero. Like I said, this is very naïve and just a suggestion. Outside of eigenforms, all I seem to be able to do is give rough estimates of the proportion of non-zero coefficients. They wouldn't tell you anything about coefficients at primes.

Naïve heuristics suggest that no such modular form exists, and surely no such eigenform exists, as you very well know. I thought one could use results about eigenform as an input to prove the general case, but I failed. The (first) difficulty I encountered is that for an arbitrary modular forms, it is not obvious how to relate a_p with the Hecke operator T_p.