[BlackFingolfin]

I think that's a misconception. Mathematicians are keenly aware that there are plenty examples in history were many people "believed" (hoped? expected?) that some result might be true because it would be "beautiful", but it turned out to be false. Or where numerical evidence suggested something only to turn out to be wrong in the end. And where the first counterexample only exists at huge, numerically infeasible bound (see e.g. https://en.wikipedia.org/wiki/Skewes%27s_number)

And hence a well-known suggestion is that when tackling a hard problem, is to try finding a proof on even days, and a counterexample on odd days...

And indeed, lots of people tried (and still try) to find counterexamples to, or otherwise disprove, the Riemann Hypothesis. However, there are indeed many, many results and heuristics that give a strong suggestion that the RH might be true -- far more than mere numerical results computing zeroes of the Zeta function. Of course none of them constitute a proof; but this really goes far beyond a simple hope for "beauty" in the theory.