I've always felt like there should be some relatively straightforward proofs by contradiction establishing theorems about prime distribution. Ideally they would exist for things like twin prime conjecture / infinitude of $x^2+1$ primes, but here, let's focus on a simpler case, already solved: Bertrand's conjecture.
Suppose we assert a prime gap larger than the prime it follows, i.e. $$\exists p_i \in \mathbb P : p_i < 2p_i<p_{i+1}.$$
Naïvely, I would think it should cause no end of problems, readily leading to absurd results which are easily digestible. I envision, for instance, a few simple steps from a Bertrand-breaking prime gap to demonstrating that that would imply some large even number is prime, or that there could only be finitely many primes, or some other catastrophic consequence. Since such an assertion seems so inimical to the overall structure of the primes, a problem thread to pull shouldn't be so difficult to locate.
(I know Erdős's proof of Bertrand is considered a gold standard of elegance, but that's neither here nor there, and is not an example of the the type of proof I attempted to describe above.)
So to reiterate,
What is it about the primes that tends to render straightforward contradiction proofs so hard to come by?
Why does such an impossible gap not cause things to break down in exploitable ways? Do the primes as a set "self-repair" their distribution so that the consequences of such assertions fade away over time rather than get amplified and cause cascading problems?
Any reasonably cogent explanation along these lines would be a suitable answer (even if the answer is ultimately "we don't know").