There were literally just House hearings about the impact of DoH on law enforcement investigations. You lobby against it by trying to get a law passed to ban it.
Are you sure you're not thinking of hearings about some other aspect of encryption? I've been working on the DoH issue, including writing to Congress about it, and I don't recall a hearing having happened yet.
If web browsers want to allow playing content from DRM'd services like Netflix, they get tangled up in a bundle of regulations like the DMCA and have to use specific services like this: https://en.wikipedia.org/wiki/Widevine
At a lower level, formats like H264 are basically part of the web platform too, so you end up implementing them and that exposes you to software patent laws.
So, in practice: Yes, there are laws browsers must adhere to. If you're willing to strip a bunch of features out of your browser and you live in a country with a less awful legal climate, maybe not too many laws.
For a while the US banned export of encryption if you aren't familiar with that - it impacted stuff like web browsers and resulted in people using 56-bit (!) encryption keys for things.
In practice you can host your browser redistributable on non-US server that has no obligation to US laws. And if US user chooses to download from that location there is nothing they can do about it.
It's not like Google or Mozilla were trying to write a new law.
Are there laws that browsers MUST adhere to? Because I'm pretty sure I can create a web browser that behaves in any way that I see fit.