If it's WWIII, and you're using ballistic missiles against satellite constellations, then either:

- You are not targeting individual satellites; you're setting off nuclear warheads in space, and relying on the EMP to disable all satellites within a large radius of the blast - https://en.wikipedia.org/wiki/Nuclear_electromagnetic_pulse

or

- You're nuking the ground-based command & control centers for those satellites. Again, nothing like 10,000 missiles needed.

(Or both.)

To target 10,000 satellites directly, the "obvious" weapon would be a few satellite-launch rockets, lofting tons of BB's (or little steel bolts, or whatever) - which would become a sort of long-duration artillery barrage shrapnel in orbit.