I agree with you that the game shouldn't auto count down.. Instead it could count down a separate 10 sec, incase you abort. On the other point, the games should end when the arena ends.. This has been debated before in great length, and the majority of the people want the games to end when the arena ends..
Tournament Arena Issues
I am aware that both options have their up and down sides. Sure it gives a few seconds to decide which opening you want to play. On the other hand you don't know when you're going to connect and that you'll be playing white and it throws you into the game with basically a few seconds short which can be a deciding factor. In non tournament games I feel like there is a few seconds before the clock starts ticking. Black could be given the same time to react to on the first move to make it fair for both sides.
Yeah, there's a one or two second delay for white. I think black also gets a second or so, but I can't remember.
This one might be a bug and not intended: After each game there is this overlay showing who won etc. Once the time runs out, the current games stops. When you then join a new tournament, this overlay reappears offering to go to the next match, while a new opponent is being searched.
I feel like the last game shouldn't stop when the tournament time runs out. Maybe players could toggle a continue option and if both have it on, the game will continue, just not count towards the arena points. The results shouldn't be displayed while an arena or non-arena game is ongoing.
It seems like in the tournament, e.g. 1m bullet, the time starts running immediately instead of after the first move. Is this intended? It sort of puts white in a disadvantage as you're dropped into a game and don't know you'll have white and the time is already running.
Lastly. I might be remembering it wrong, but I think there was a "cancel" button where you could abort the search for a new opponent before you're connected, it seems to not be around anymore. I don't mean canceling a game before making the first move, I mean canceling the looking for a new opponent. If you close the overlay with the next match button, it's still searching. I think it's not in regular live chess either.
Edit: Clicking the searching in progress box actually aborts it. Maybe it could be a bit more visible that it has that function, not only on mouseover, especially since there is non on mobile.
Also it's missing the buzzing sounds for illegal moves like when a piece is pinned to the king or trying to castle when you can't. It's not always clear if one dragged it on a wrong square while hurrying or if the move was illegal. Well, that would also be an illegal move, but in the sense of that the piece could go there, except you're in check or pinned.