Cloudflare at least thinks exploiting from a browser is possible, but I haven't seen anyone claim to have done it. I suspect if it is possible it isn't as efficient as the assembly PoC. I read some people saying that running the PoC code for 10 minutes they already saw fragments of their root password leaked.
I actually think this is pretty horrible and AMD rating it 'medium' severity is a joke. Haven't seen any info on performance hit (if any) from the updated microcode. I'm not sure why it will take so long to get the patches out for the rest of the product stack. This is serious enough I'd think releasing those immediately with only that fix would be wise. Certainly it is game over territory for cloud providers, but it doesn't seem an overblown worry for regular users at all to me.
I haven't seen anything to suggest other zen generation products aren't possibly vulnerable, but the language used could just be because you can't easily prove a negative not because they haven't tested. I guess we'll know soon enough.
The only good news is the chicken bit mitigates it.