I'd consider it much less likely than e.g. nuclear or maybe chemical/biological warfare.
Kessler syndrome (if even achievable with current technology) would be a major bummer for science and the global economy for a couple of decades (no more Starlink, but we still have good old geostationary satellites, so no ships and airplanes would get disconnected as a result), or at worst centuries, but would otherwise not form any threat to civilization, whereas nuclear winter is already very capable of wiping it out.
> if even achievable with current technology
Launching a nail bomb into orbit would've been possible as soon as we were able to get into space, the only question is motivation. A terrorist state, say North Korea, threaten the rest of the planet and demand concessions once they're able to get any significant mass into orbit.
I think it would be actually the other way around - Starlink orbits aee low enough to be self cleaning & Starlink satellites can be (and are) rapidly replenished. So even if something from up above hits anfew, the debris would deorbit soon & new ones could be launched.
With GEO sats, unless you go for direct GEO insertion, it might still have issues reaching the final orbit. And even at GEO, there could be a debris cloud as well causing issues, at least until the sun and moon gravity perturbs it enough.