No. First off, these satellites are all in the relatively underpopulated upper section of low Earth orbit hugging the lower reaches of the inner Van Allen belt at ~1,000km above the surface. Most satellites in LEO are closer to 300 km up. So there aren't really any non-ElonNet satellites to worry about except for ones in highly eccentric polar orbits and those are few enough to be avoidable. They also have fairly aggressive plans to de-orbit them at the end of their lives which makes me happy.

The big danger for Kessler syndrome right now is all the satellites whizzing around Earth in polar orbits that all cross over the Earth's poles.