In a word, yes. But few spacecraft have objectives that require them to be in that region, so they avoid it if possible, mostly for the reason @Hobbes already stated: radiation. But there are missions that require a spacecraft to operate there. One example is NASA's Magnetospheric Multiscale mission (see https://en.wikipedia.org/wiki/Magnetospheric_Multiscale_Mission), with a perigee of 2550 km. If you know you'll be operating there you design for the radiation with rad-hard avionics, materials that can handle the radiation with no or acceptable degradation over the mission lifetime, etc., but that increases the cost of developing the spacecraft. Spacecraft whose missions are in LEO or GEO generally don't spend the money to build for MEO tolerance, so when they go to MEO for disposal they don't last long.
As far as the "space junk" collision hazard, as @JCRM noted it's a big region. I calculated the volume of that region, including all orbit inclinations, so it's a sphere with a smaller sphere removed from the center: ~6.4 x 10^13 cubic kilometers! The most heavily populated LEO region, from about 200 to 800 km altitude, has a volume of ~3.6 x 10^11 cubic km, more than two orders of magnitude smaller than the disposal region. For now the collision hazard is significantly less in the disposal region than in LEO. In the coming decades, if they start sending hordes of CubeSats into there (after their high-LEO missions are finished, of course) that might change.
One last issue with operating LEO-intended satellites in the disposal region: telecom. The telecom system for an LEO satellite will have been designed for the required data rates at the distances it would see in LEO. Overdesign costs money! When you boost the satellite into MEO those distances increase so the supportable data rates decrease, roughly as 1/r^2 (as long as the indicated data rate is ~1 kbps or more). If the extended mission can live with the lower data rates, and the satellite's telecom system design supports them, then yes, you could operate such an extended mission in the disposal region, until the radiation kills it. That is, IF the radiation environment doesn't cause too many bit-flip errors, which can have a huge range of unpleasant effects, from corrupted data to power distribution system controllers and breakers that simply quit working.