Firstly, when you perform multiple hypothesis tests (as you do by looking at whether p-values from multiple outcomes), there is in principle a multiplicity issue in the sense that comparing each p-value versus the level $\alpha$ will result in a familywise type I error rate $>\alpha$. I do not think this really goes away, if you do cross-validation. Whether you need to control the familywise type I error rate and across which analyses is a complicated issue. E.g. if you write two separate papers on the same dataset, do you get twice the familywise type I error rate, but not if you put the results in the same paper? This is really only (relatively) clear in a few settings such as confirmatory clinical trials for getting regulatory approval for a drug.
Secondly, the practical reason why many people are keen on adjustments is, because many people take a "many shots on target" approach, where they do lots of comparisons and then emphasize those with an unadjusted p-value <0.05. It is clear that when people study a lot of things including a huge number of things that really do not affect the outcomes being studied, that this will fill the scientific literature with many purported findings that are just random noise. This only gets worse when there are many small decisions left open until the data has been collected, which may lead to the potential for the choices in the analyses being data dependent. It may be debatable whether multiplicity adjustments help that much for such issues, but I guess I am not alone in trusting results with p<0.05 less when (a) the study was not pre-registered with outcomes and analyses pre-specified, (b) lots of outcomes were studied, (c) no adjustment was made for multiplicity and of course (d) the claimed effect is not a-priori plausible.
Thirdly, you should not completely change your view of results just because p=0.02 or p is just over 0.05. The former is not completely compelling evidence (and I would not get too excited about it) and the latter does not mean that the hypothesized effect is not there. Of course, this may affect what journal editors and reviewers will let you write (and whether they get excited about your paper) so in practical terms it may sadly be a major difference.