I'm a bit more hopeful about Mozilla. I think there would be a collapse of the business without the Google cash flow, but expect the browser itself would be continued by the community for its importance.

I have about as much faith in the open source community maintaining a machine as complicated as a full browser without significant corporate backing as I have in OpenOffice displacing Microsoft Office in my lifetime, but hope springs eternal.

Consider that KHTML, the grandfather of just about every modern engine that isn't Gecko, was a fairly small open source project. Ladybird is open source and surprisingly usable for its youth and single developer.

Speaking from personal experience, I write my best code when I'm frustrated by the absence of something I need. I'm sure many others here are the same way. If a sufficient fire is lit under the community to make the web continue to work in the absence of the old guard, I'm pretty sure it'll happen.