Not correct. In both Taiwan and SCS they are claiming things that are not in their possession. Here they have already secured everything 50 years back. Now, The disagreement here is on minor pieces of land. Which is inconsequential in their larger scheme of things. (Also, the discussion you quoted was about PRC going all-in for this)
Where is this 'tibet or xinjiang breaking away' coming from? Ürümqi is 1200km away from Ladakh. In Tibet, we do not have any leverage (Dalai lama's time is limited). A local conflict in aksai chin is gonna blow over to these places? hardly so. If sea access is lost then next priorities would be securing energy access to central Asia and east asian borders before they look to Tibet. Simply because of the population distribution.
The Karakoram is not the only land access route to the outside. Its just a highway, they have rail access through multiple central Asian stan countries. Dont know how many highways. Karakoram is special only because of it Indian ocean access. It alone simply cannot give its mainland any kind of resource security.
And cutting off Karakoram High way is very easy. It has many bridges build at very high altitude. A handful of cruise missiles can blow up it and disabling it for very long time.