Dove is a suite of configurations & advanced modifications for Mozilla Thunderbird, designed to put the user first - with a focus on privacy, security, freedom, & usability.
Librewolf is a privacy oriented fork of Firefox, it grabs some setting from arkenfox. Betterbird is not a privacy oriented fork of Thunderbird as far as I remember. When I tried it the only thing I was attracted to was its tray support, but as I use non DE compositors, so far wayfire, labwc and sway (tabbed layout), and as there's currently a Firefox bug, I didn't see any reason to keep trying it, and now on sway with tabbed layout I see no reason for a tray any ways...
Did you read their website/faq? It very clearly tells me that Betterbird has a very different focus than LibreWolf. It's not about Privacy, it's about fixing bugs.
K-9 mail is literally Thunderbird. It’s been rebranded and taken over by Mozilla. They’re keeping the k-9 branding as its own (otherwise identical) app as a nostalgia token for the people that have used it for a long time.
There are no "problems" with Firefox. The problems are with Mozilla and how they operate Firefox, so they could easily start affecting Thunderbird too.
I tried to use LibreWolf yesterday. The Flatpak version wasn't supported by KeePassXC, so that was a pass for me. The Fedora version did work with it, but it was also freezing my system and causing graphical glitches that were so terrible, I thought my hardware died. There were coloured blocks as well as weird pixelated warping. It's probably because I have Wayland, and I noticed that the Fedora version ran under X. I would have submitted a bug report, but I didn't even know where to start. Maybe I'll do that eventually.
E: I have narrowed this down with journalctl to be an issue with amdgpu. This happenes on Firefox as well, but not other applications somehow. It's something to do with a gfxhub page fault. I'll have to see whether an older kernel version might help.