Nah. In any major future conflict, the combatants will go ahead and launch replacement satellites immediately regardless of the risks or long-term consequences (or they'll do it at least as long as their manufacturing and launch facilities survive). A constellation of hundreds of satellites can't go "boom" all at once. Even with a bunch of orbital debris floating around the hazards will be sparse and some satellites will live long enough to be operationally useful.

For the purposes of the crisis, sure. But commerce and average consumer internet access will suffer hugely. Similarly, severing the sea cable had no direct military effect, but was economic damage. Kessler syndrome is still a serious concern even in LEO, just not to the same extent of practically denying access to space for the foreseeable future.