Observationally, the vast majority of mathematical papers do not make claims that are non-rigorous but as well supported as the Basel problem. They split into rigorous proofs (potentially conditional on known additional hypotheses eg. Riemann), or they offer purely heuristic arguments with substantially less support.

It should also be noted that Euler was working at a time when it was widely known that the behaviour of infinite sums, products and infinitesimal analysis (following Newton or Leibnitz) was without any firm foundation. So analysis of these objects at that time was generally flanked with "sanity check" demonstrations that the precise objects being analysed did not trivially cause bad behaviour. Essentially everyone treated these kinds of demonstrations as highly suspect until the 1830's and a firm foundation for analysis (cf. Weierstrass and Riemann). Today we grandfather these demonstrations in as proofs because we can show proper behaviour of these objects.

On the other hand, there were a great many statements made at that time which later turned out to be false, or require additional technical assumptions once we understood analysis, as distinct from an calculus of infinitesimals. The most salient to me would be Cauchy's 1821 "proof" that the pointwise limit of continuous functions is continuous; counterexamples were not constructed until 1826 (by which time functions were better understood) and it took until 1853 for the actual conditions (uniform continuity) to be developed properly. This statement was at least as well supported in 1821 as Euler's was in 1735.

As to confidence in modern results: Looking at the Web of Science data collated here for retractions in mathematical fields suggests that around 0.15% of current papers are retracted.