EDIT: I've been getting a lot of down votes for this stance, surprisingly. Why not share your position if you don't agree? There must be a bunch of hardcore people on here who are writing directly on the metal in machine code. Short of that, you are in fact using someone else's code in all your projects.
I think that when most devs hear "using existing code", they're thinking of the latter, not the former.
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Will they though? Maybe some toy ones but the distribution of other browsers are largely built on chromium with a few on Firefox and one or two on WebKit.
Beyond that, there’s Arc that’s made a splash with the HN crowd but IIRC, it’s VC funded so a whole different set of concerns.
There will be other browsers but almost the entire browser market is essentially funded by Google.