Bayesian methods are better in a number of ways, but ignorant people using a better tool won't necessarily get better results. I don't think the net effect of a mass switch to Bayesian methods would be negative, but I do think it'd be very small unless it involved raising the general statistical competence of scientists.

Even when Bayesian methods get so commonplace that they could be used just by pushing a button in SPSS, researchers will still have many tricks at their disposal to skew their conclusions. Not bothering to publish contrary data, only publishing subgroup analyses that show a desired result, ruling out inconvenient data points as "outliers", wilful misinterpretation of past work, failing to correct for doing multiple statistical tests (and this can be an issue with Bayesian t-tests, like those in the Wagenmakers et al. reanalysis lukeprog linked above), and so on.