not yet rolling out 
but I’m currently preparing for rollout, should happen tomorrow. (decided to delay everything a bit due to some bugs and cool features)
It is indeed working, I was offered both 3.333 and 3.335 automatically now.
A minor quibble would be that with BTT's Dock and menubar icons both disabled, the only thing that (visibly) happened was that the Dock icon appeared, in the background. Only when I clicked it did the actual "update available" window appear.
Maybe you could issue a notification (macOS or custom) when an update is available? Or at least make the update window take focus or let the Dock icon bounce a few times; otherwise this is very easily overlooked or even mistaken for erratic behavior.
Hehe, there has been a LOT of feedback over the years that people don't want anything which steals the focus from their work. Which is why BTT has switched to the background update notifications a while ago.
But maybe I can add a notification option
Yeah, I can imagine. 
But that's the kind of thing native notifications are great for, since you can configure them any way you like—from outright alert to never ever seeing them.
If only people knew about that (also had notifications for quite a while in BTT, but apparently almost nobody knows you can disable them per app). Maybe this has changed a bit on Catalina.
That is surprising. Indeed best if it's an option within BTT as well, then.
In my Mac (Catalina 10.15.3) BetterTouchTool only honor the Preferences's "Launch BetterTouchTool on startup" (and creates the Login Item) when it's installed in /Applications. If it's installed in ~/Applications, "Launch BetterTouchTool on startup" is NOT honored (nor Login Item created).
Please, can this be corrected? Thanks.
Hey, I had been getting this error and these eronious startup entries as well (M1 MBP, 12.4 / BTT 3.786). Used AppDelete to remove BTT and all it's support files (remember to export your settings!) and when reinstalled as if new, the setup process offered to move the app to a non-read-only folder and put it in ~/Applications/
Which is weird but it works as expected now so ¯_(ツ)_/¯
Perhaps move the App to /Users/[your usernam]/Applications/ and see what happens...
Hope this helps someone
I've been using BTT for maybe 4 years, pretty sure the auto-updates have never worked for me.
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As per what I and @Andreas_Hegenberg wrote above, I too thought I had that issue because I didn't realize updates might be offered when manually checking but not (yet) when auto-checking (which surely is confusing for users, but that's how Andreas handles it). Once I just stopped checking manually (unless I had a specific reason to do so), it turned out updates were indeed auto-checked and offered properly. And note sometimes there might really be no "official" new version for weeks or even months.