How do competing browsers make money? Chrome can do the same. It may mean users start paying for it, or ad supported in some way, which will be annoying, but even out the playing field.
The fact that Chrome is free is the problem. The fact they have such a leg up on everyone else is the problem.
Does a browser need to make money? Chrome is mostly open source as far as I know. Maybe it could move to a model like Linux where you have a benevolent dictator overseeing things and if a company wants something supported in chrome they can fund a developer to make the work happen.
As far as security and support perhaps companies like Microsoft which use parts of chrome as basis for Edge could employ people who can fix issues upstream.
Chromium/Chrome can also get ChromeOS and it's hardware division since most of the complications in chrome are there to reimplement a Phone/Desktop OS.