Not if the company providing the competing browser goes out of business and stops offering it.
That's really what antitrust regulation is for: to prevent one dominant company from killing all their competitors. Once that happens, the dominant company has no more pressure to succeed and their product stagnates, harming everyone.
Browsers are actually a great example of this; despite the government prosecution, Microsoft succeeded in killing off Netscape and then had the entire browser market to themselves. The result was IE6 and "made for Internet Explorer", a horrible mess that took years to clean up.
But it’s not really how the world works though as bigger companies use this soft power for everything, which is why I have stocks in FANG and it just keeps going up and up.
It’s a bit like Atlassian, no one is happy with Jira by itself but they do have the whole package.
https://slate.com/technology/2009/05/google-s-first-ever-tv-...
The “try Chrome” ads for other browsers didn’t start for a while later after that IIRC.
I think you are having a very selective memory. Since day one, Google put Chrome ads everywhere including on the Google home page when you are visiting from Firefox and there is no way to permanently disable them.
If you are aware of a similiar marketing outreach that Mozilla could do I'm sure they would love to hear about it!