This is likely overly optimistic, but can one take the results from the Modularity theorem and Ribet's theorem, and distill these down to an undergrad math level of a way to check if certain rational polynomials have no non-trivial solutions?
For instance, I do not understand the details of either theorem, but if I can reduce any equation to implying solutions of the form $$a^n + b^n = c^n$$ for integer $n>2$, I can use Fermat's Last Theorem to prove the equation has no non-trivial rational solutions. What I'm curious about is whether, even if in only restricted cases, a procedure can be formed from the Modularity theorem and Ribet's theorem to give a relatively straightforward procedure to check if certain rational polynomials have no non-trivial solutions.
I've also seen claims that one can prove $$a^n + b^n = 2 c^n$$ has no rational solutions either, with aid of the Modularity Theorem. http://webusers.imj-prg.fr/~loic.merel/winding.pdf
So I'm wondering if given a rational polynomial function $f(x)$ if there is some procedure to check if equations of the form:
$$k_1 f(a) + k_2 f(b) = k_3 f(c) $$
(where $k_1,k_2,k_3$ are some given rational numbers, and $a,b,c$ are positive rational variables which we are trying to find values satisfying the equation)
have non-trivial solutions?
What makes Fermat's equation really hard is the presence of one of the unknown variables, n, as an exponent. This makes it an example of an "exponential Diophantine equation". If one has a "polynomial Diophantine equation", (polynomial function in some number of variables) = 0, then finding rational solutions is still rather difficult, but not usually quite as fiendish as Fermat! So the class of examples you describe in your last paragraph is rather different in nature from Fermat's problem.
That said, if you are interested in seeing what else one can do for "Fermat-style" equations using a combination of modularity theorems and other techniques, I strongly recommend you browse through some of the work of my colleague Samir Siksek (http://homepages.warwick.ac.uk/~maseap/), who has proved some cool theorems along these lines (these lecture notes might be a good place to start).