Once you're dealing with real people, early termination is a huge deal, and it quickly goes beyond just the interpretation of statistics. You almost always want to have 3rd party referees involved anyways.
The reasoning for doing so is much more coherent (imo) in a Bayesian framework where you don't have to say that you are "going beyond" statistics.
Some of the frequentist approaches to early-stopping seem pretty coherent to me. Curtailiment, for example, just stops collecting data once it will no longer change the outcome of a test. The frequentist emphasis on error control for the procedure also seems like a reasonable fit for a regulatory regime where you're actually making decisions (no argument that it is...weirder for basic research)
At any rate, some approaches for early-stopping have sensible frequentist and Bayesian properties, which is nice.
Agreed that things are weirder for basic research.
A family member was part of the control group for a cancer treatment study a few years back. The study chose a stopping criterion based on Bayesian methods. Relatively early into the study they were able to determine that it made sense to move people from the control group to the treatment group.