I wonder if there's a limit to space junk beyond which leaving the Earth in a space shuttle becomes impossible.

It is already impossible - all the remaining Space Shuttles are in a museum, not to mention all Space Shuttle missions were (and were always intended to be) to Earth orbit. No Space Shuttle ever went past 600 km hight Earth orbit.

These satellites are low Earth orbit (LEO)

They're extremely sparse. Imagine putting 12,000 satellites randomly over the surface of the Earth. You're just not going to bump into one, statistically. Now expand that into 3D space in an orbital zone above us.

It's not a collision risk.

> wonder if there's a limit to space junk beyond which leaving the Earth in a space shuttle becomes impossible

There is. We don't have the industrial capacity, as a species, to do it.