At the final analysis, neither countries want this. But it isn't what the leaders want that's the problem.
Beijing I'm fairly confident didn't start this or wanting to prolong it. I also fairly confident it has its troop under full control.
Unfortunately, whole leh saga was started by PLA who doesn't follow their political leadership. It was china who started it all in galwan and refused the implement the agreement of their leadership.
India on the other hand. Well I'm fairly certain Modie doesn't want it either (But secretly hoping to spin a win out from it). But lack total control from all the nationalists under him. And the hot head ill-discipline sodiers. As demonstrated from the dead colonel' s antics.
you rightly said , Modi doesn't want a war but that doesn't mean anyone can come and alter the status of our boundaries. As for soldiers goes, you need to see our history and discipline with which they act. Ours is the most professional and disciplined army who have never disobeyed the political leadership while it's the war with china or pakistan or liberation o bangladesh or UN peacekeeping mission, they have acted with utmost professionalism.
If they were indisciplined, they would have grounded the pakistan many times by now since their proxy war is going on for almost 3 decades now. Even in Kargil war also, when whole world knew that's it's pakistan army regulars ( 25 NLI soldiers) who have captured kargil heights, we choose to not cross the line. Infact, we gave their soldiers a respectful burial when their own country refused to own them.