Skylab was in very good condition and NASA wanted to use it in conjunction with the shuttle, which was scheduled to be operational before Skylab fell into the atmosphere. The Shuttle was to be used to re-boost it, but two things happened:  solar activity was higher than expected (which affects the upper-most part of the atmosphere and increased the atmospheric drag on Skylab) and  the shuttle ended-up being too far behind schedule. NASA, realizing that shuttles would not be ready in time, studied launching an unmanned "tug" to dock with and re-boost Skylab so it would still be there on orbit and operational by the time shuttles were ready, but congress in the late 70's was as stupid as today - Congress did not fund this cheap solution, so we ended-up dumping $100 Billion and ten years of construction time into building ISS to get a similar orbital capability (Skylab had 320 cubic meters pressurized volume, that's more than the US part of the ISS). The shuttle could have then flown additions to Skylab (which had a docking adapter for multiple visiting vehicles). An enhanced Skylab would have had no Russian "entanglements", and had its own lifesupport and navigation capabilities.
Skylab was FAR from "worn out" and the damage from the launch was quite managable. The astronauts who closed it out left it ready for re-manning. When Skylab re-entered the atmosphere it did so under remote control from the ground, with its systems fully functioning until they were destroyed by the reentry. READ THE DAMNED REPORTS, which consist of hundreds of paged of excellent details, before misinforming people.
Skylab was put into orbit in one launch. Using the shuttle to lift further components is silly- all it does is justify the shuttle. The shuttle is a Honda Fit compared to the 18-wheeler Saturn 5. Letting Skylab burn up may have been a "waste" but if you can launch more than 1/3 of the current ISS volume (currently at around 837 pressurized cubic meters) with 1 Saturn 5 rocket, an orbiting space station then becomes essentially disposable. Just launch another one.