Hello,
unfortunately, for me, this changed nothing.
Just to be clear on where the issue is, it doesn't matter if it is a game or not. Even if you write a longer chat message, and while it's being read another one arrives, the second one will always interrupt the first one. Of course, it depends on how fast your screen reader is and how much will be read in a given game.
One workaround I have noticed is that if you use page up and down to review the history, while a message is being read that way, incoming new messages don't interrupt it. Changing game messages to high or critical doesn't solve anything though, because now again everything interrupts each other as before, with an additional sound playing (VoiceOver on the Mac plays a sound for aria-live=assertive messages).
For your other question, yes, Chrome and Firefox both work with VoiceOver, with various advantages and disadvantages, and the same is true for the Playroom web client. I could go into detail regarding these, but for this particular issue, everything is the same across Firefox and Safari, Chrome is honestly the worst here, but I doubt that's a problem on your side. In Chrome, if, for example, 3 messages arrive quickly in a game, the reading order is totally inconsistent with no way to explain that easily. Sometimes maybe they will be read in the format 2, 1, 3, the other times they will be read normally, another time in the format 1, 3, 2, and so on. These numbers represent the number of the message arriving, e.g. first message is 1, second message is 2 and third message is 3.
Chrome doesn't seem to interrupt messages but it's quite pointless when they are being read in this inconsistent way and it's very difficult to follow what is going on, especially since there are more often much more than 3 messages arriving, for example if in a Duck racing game I skip 2 turns and the bot is playing, the order of what VoiceOver is reading completely makes no sense.
Well, to finish up this long post, I can test any other change made, or answer any additional questions you may have.
PS: I noticed you enabled musical / counting duels on the web client as well. Was that intentional? Counting duels seem to work fine, but musical duels seem completely broken. No notes play, whether if I press the numbers or if they are supposed to play in a sequence.