How up to date the embedded web engine is (and how quickly you would be able to merge upstream changes, update the framework, wait for "browsers" implemented using the embedded framework to update their dependencies, etc.) That can be a pretty big deal when there's a fix for a high-impact vulnerability.
With a known browser, you get reasonable guarantees that TLS is being handled properly. With an unknown browser, these guarantees are gone. Considering the average person has no idea what https/TLS is, banning all unknown browsers as a defense against phishing seems completely reasonable.
For the most part, if a browser's extension store isn't in order, everything else they claim to do to secure web traffic is kinda a joke.
As for your examples, Konqueror and Falkon might be blocked because they use Qt WebEngine – which embeds Blink in the Qt framework [0].
[0]: https://en.wikipedia.org/wiki/Blink_(browser_engine)#Framewo...
In this case it's blocking the embedded Blink, which is their browser engine, so at least they're consistent?
Blink forked from Webkit, which started as a fork of the KHTML and KJS libraries from KDE. Many KDE-projects are built upon Qt, but Qt is an independent project.