If you don’t want to use Gecko nor Chromium, I am aware of the following alternatives:
WebKit
Though associated with Apple and Safari, WebKit (@webkit@front-end.social) has its origins in KDE and its Konqueror browser. KDE developed its own web engine called KHTML, which was forked into WebKit. It’s therefore fully open source, despite the Apple connection.
On Linux you can use WebKit in GNOME Web (formerly Epiphany) or Konqueror. If you’re on Mac, Safari is probably your best bet. Windows users appear to be out of luck.
Servo
Servo (@servo@floss.social) is a brand new Rust-based engine which was originally developed by Mozilla, but which was abandoned by them like good things often are. Thankfully the Linux foundation took over developments. It’s still in development, but from their download page you can take it for a spin within seconds on all three major operating systems. It’s looking pretty good.
They maintain a list of things made with Servo. The most promising project so far appears to be a browser named Verso.
Ladybird
Ladybird is another development to follow. Unlike WebKit and Servo, Ladybird is being developed as a web browser in its own right, but this browser will come with a completely original rendering engine. It aims to have an alpha released next year, and is largely written in C++.
I’ve been inching towards Gnome Web (WebKit) for a while now. Every time I try using it I last a little longer than last time before I encounter some deal breaking issue and return to Firefox.
In the short term I’m considering sticking to Firefox for work, and using Gnome Web for all other kinds of distractions. I’m writing this in Gnome Web right now, and it’s working great. :)
On mobile I’ll probably stick to Firefox for a while. So they will still have all my data, and if I have to choose between Mozilla and Google it will still be Mozilla. But my god I wish they would stop acting like idiots.