Even though you can analyze a test Bayesianly, you still need to know about power (or some Bayesian alternative to power) to plan a sample size.
Especially with proper priors (which again, everyone running an A/B test should have at this point) you don't have to "plan a sample size". You work with the data you have, and assess the risks in the decision give what you know. There is likewise no risk of "early stopping".
If you're doing Bayesian analysis and only have say 100 visitors, and are using priors, the only situation in which you'll have a strong posterior is when the winning variant is notably superior to the current one. All of the ad hoc rules Frequentists put in place are not necessary when you do proper analysis of your posterior probability that A > B.
Again we see where Bayesian vs Frequentist doesn't really matter for a clinical trials but does for running an A/B test. In a clinical trial you are going to have to plan how many people are involved anyway, so doing a test power calculation makes sense since you'll need some number and that's a good way to get one.
For running an A/B test the bigger problem is usually time not numbers (an exception being email campaigns in which both tend to matter). In the Bayesian setup you if your boss says "I need an answer tomorrow morning" you can take the data you've got, show the probabilities of improvement as well as the distribution of the risks if you're wrong. This allows you to make riskier moves when it makes sense, and be more conservative when it does not. This is something that the out-of-the-box NHST does not allow you.
Even a Bayesian with a proper prior can make good guesses about sample sizes, for example, by saying that they have a goal to reduce the length of the 95% credible interval for the most relevant parameter by 80%.
* While you don't have to have a fixed sample size up front, you can still "cheat" in a bayesian analysis if you peek constantly and end early on promising results that you want to win, and let them run longer otherwise. So you want to do something to account for this (put some structure in place, approach with skepticism, laugh and put on sunglasses, whatever).
* It's very often useful in practice to have some idea of what kind of answer you're going to see in how long for planning reasons -- for example, rather than your boss saying "I need an answer tomorrow" they say "I need an answer as quick as you can". Bayesian methods give you the flexibility to be risky when you need to and accurately count for uncertainty, but sometimes you still need to predict and strategize around ideas like "We'll be about this certain in 2 days, and about this certain in 1 week, and about this certain in 4 weeks and it seems like planning on next Tuesday is the right call"
I've found understanding these frequentist methods to help inform my guesstimates of how experiments will play out with regards to sample size and impact as well as honestly evaluate the trade-offs in evaluating the tests where I wasn't running it -- AB testing is really widespread so I feel like it's important to understand frequentist tests well even if you intend to never use them if you can help it.