graphite-or-graphene BadRobot
Sorry but first of all, what is soo bad about the default keyboard? It is perfectly functional as a default keyboard.
Also, there are other options other than GBoard if you want. I personally like HeliBoard. It is another open source keyboard based on AOSP OpenBoard, but it has many great/useful features. One of the app variants even allows you to load a gesture/swipe typing library, if you want that feature.
Last point though - I have disabled all other keyboards and spellcheckers, other than the GrapheneOS Keyboard and spellchecker, so I could test your situation, and my autocorrect and spellchecker is working just fine in my browser (Vanadium). I have tested it on multiple websites as well.
Perhaps you have something configured in a way that prevents yours from working? Have you made sure your keyboard and spellcheck settings are turned on and everything?
Apologies if I have misunderstood something by the way, but yea everything is working fine on my end so just thought I would comment in case I could help at all...