Inadmissible theorems in research
Does maths research have anything inadmissible?
No, but trying to prove X without using Y is still a very useful concept even in research, because it can lead to interesting generalizations, or new proof techniques that can be applied to a larger set of problems.
For instance, in some sense the Lebesgue integral is "just" trying to prove the properties of integrals without using the continuity of f, or the theory of matroids is "just" trying to prove the properties of linearly independent vectors without using a lot of properties from the vector space structure.
So this is far from being a pointless exercise, if that's what you had in mind.
In the sense that you are asking, I cannot imagine there ever being a method that is ruled inadmissible because the researcher is "not ready for it." Every intellectual approach is potentially fair game.
If the specific goal of a work is to find an alternate approach to establishing something, however, it could well be the case that one or more prior methods are ruled out of scope, as it would assume the result that you want to establish by another independent path. For example, the constant e has been derived in multiple ways.
Finally, once you step outside of pure theory and into experimental work, one must also consider the ethics of an experimental method. Many potential approaches are considered inadmissible due to the objectionable nature of the experiment. In extreme cases, such Nazi medical experiments, even referencing the prior work may be considered inadmissible.
It is worth pointing out, that theorems are usually inadmissible if they lead to circular theorem proving. If you study math you learn how mathematical theories are built lemma by lemma and theorem by theorem. These theorems and their dependencies form a directed acyclic graph (DAG).
If you are asked to reproduce the proof of a certain theorem and you use a "later" result, this results usually depends on the theorem you are supposed to prove, so using it is not just inadmissible for educational reasons, it actually would lead to an incorrect proof in the context of the DAG.
In that sense there cannot be any inadmissible theorems in research, because research usually consists of proving the "latest" theorems. However, if you publish a shorter, more elegant or more beautiful proof of a known result, you might have to look out for inadmissible theorems again.