I would call that feature the "sandbox", where you can set up your own positions and go over them with your opponents. You can play untimes games with them, but you don't have to play the whole thing.
New option required

There are two clocks, one for each player, and only one counts down at a time. Your clock counts down when it's your move, and your opponent's when it's their move. You can only lose on time when it's your clock that runs out.

"suddenly informed......."
eh ?
if your keeping an eye on your remaining time while you play , as you should , then it can hardly be a suprise when your clock runs out.
and if youre significantly advantaged with either material or position but massively behind on time then look for a draw with repeated check or some other stalemate producing tactic.
dont really see any fault with the system
Live chess is fine, but I am going to stop using it unless there is an option added that allows you to play out the whole game!
As it stands, you have two opponents, the other player and the clock! It is impossible to make a long term plan because regardless of how good your position may be, when the time runs out the software seems to arbitrarily select one side or the other as the winner! It is very frustrating if you are in the last stages of planning your attack and are suddenly informed that your opponent is the winner because the clock has run out!