I think we might be misunderstanding each other a bit.
First of all, none of my posts were talking about a specific browser (and, as I understand it, Google's proposal is supposed to apply to all browsers). You've mentioned Chrome, but you're the first to mention it (or, indeed, any browser) by name in this discussion. As you say, Chrome itself isn't open source and therefore its restrictions are not entirely relevant here. Obviously we can't ignore it completely in the overall discussion about Google's proposal, but in the context of "checking" open source browsers we should be able to put it aside.
Some browsers do allow you to modify them as you see fit. For example, Gnome Web is relatively popular on Linux and it's covered under
GPL, and my understanding is that this more-or-less says "you may modify and redistribute this, as long as your redistributed/modified versions stick to this same licence". I believe Firefox is similar.
You've mentioned certificates for authentication, but I'm not sure how this would work. Using a "true" open source browser as an example:
Scenario 1. When you download the source code, it comes with the required certificate. You build your own version of the browser, then you sign it using that certificate. Your modified version would therefore pass the certificate check, and at that point what's the purpose of having the certificate in the first place? Yes, it could work around "hacking" the binary, but is anyone actually going to do that if they have the original source?
Scenario 2. The source does not come with a certificate, and you need to get your own and sign it yourself. It will now, presumably, fail the certificate check. Yet the browser hasn't been "tampered" with: the changes that you've made are expressly permitted under the software licence.
There's probably a third scenario that I haven't thought of