I've been on the Google Play version of K9 for years and I must say that it was perfectly fine. I changed to FairEmail as I thought having a very old version of an email client might not be ideal from a security perspective. At the moment the biggest change from my workflow is the UI, which looks better that the old K9. Now I am not switching to K9 F-Droid version just because I am lazy to configure everything again another time.
But please bear in mind that my way of using email on my phone is very minimal. I usually just need something to read and for most of my productivity task I use mutt.
We had a HN thread a few weeks back on the K-9 Donations blog, basically the K-9 Play version is still the old client UI (but the betas are not 100% feature complete); you can opt-in to the K-9 beta on Play or F-Droid and get the new UI experience if some features like IMAP IDLE are not that important to you. (side comment, the new K-9 betas allow export/import of settings but not sure about the older version you're currently using)
I assume they're referring to TLS client certificate authentication support. As you say, most email clients can handle TLS.
Few can handle authenticating to the server with a client certificate, but FairEmail seems to support it fine, although I've yet to configure it on my own server as it's likely to break other clients that don't support it.
If it (K9) does, I certainly did not find a way to do it. Yet, it does not mean just straight TLS connection at the start of the session. Rather, it means that when the client (FairEmail, in this case) performs the handshake, it also sends its own certificate that the server can verify prior to finishing the handshake, therefore adding an extra layer of security.
Just to emphasise, this is probably not something that most people would ever need, but is certainly an important feature to me and something that FairEmail supports and apart from a rare few, nobody else does.