The Subtle Art of the Mathematical Conjecture

Yes, exactly. And more importantly, resolving mathematical conjectures is more about developing new theories than it is answering singular questions. We don't just care about the Riemann hypothesis because of its utility in solving problems related to the first n numbers; we care about Riemann because of how its implications can be used to solve problems concerning number systems in the abstract.

It only seems strange to continue doubting a conjecture like Riemann if your perspective on the subject is experimental. In the scheme of things the relative impact of Riemann on the first trillion cases isn't nearly as important as what it says about all numbers.

10 trillions is not much in the realm of "conjectures that have only large counterexamples". The proposition

𝑛^17+9 and (𝑛+1)^17+9 are relatively prime

is not true for all n, the first counterexample being n=8424432925592889329288197322308900672459420460792433

Source: https://math.stackexchange.com/a/365881 which has other nice examples