It's actually thought that Kessler syndrome is kind of already happening right now, which is why there's a lot of push right now to try to de-orbit the very large pieces of debris, so they can't act to form further debris.
The reason it's a scary outcome is because it's an exponential. It can look like an isolated incident or incidents, then the next day be not practically stoppable.
And I'm telling you, your "image" of what this looks like is just incorrect. The kessler syndrome is likely already occurring. Yes creating more debris will make it happen more, but it's not like lighting a match to a pile of tinder.
And it's not in fact exponential in the sense that people commonly imagine when they hear that. It's an exponential that's very close to flat, i.e. an exponential with an exponent barely above one. Given enough time, yes it can destroy all satellites in Starlink's orbit, but it's not on time scales that's relevant to a war.