What I'm not seeing the answers here address is the tracking issue, which seems almost certainly part of what they are talking about.
NASA tracks US satellites (and the shuttles when they were operational) with a couple of ground stations called TDRSS. These themselves use satellites, many of which were launched by Space Shuttle missions in the 1980's and '90s.
I don't know a lot about the composition of the first TDRSS, but I worked on the system's software for the second TDRSS (STGT) from about 1989-92, so I know a bit about that system. It used a set (6ish if memory serves) of VAX clusters. Each cluster contained 2 full-blown VAX VMS mainframes, and a third small machine. One mainfraime in the cluster would be the master that was actually operating things, the other was a "hot" standby that was doing all the same work, but not transmitting, and the third's only purpose was to detect if the master crashed, and help promote the hot standby to master. Clusters themselves had hot spare clusters in case the whole cluster went down.
The point of this story though is that this wasn't really run by that many computers. Low double digits of commercial VAX machines. While these weren't entry-level VAX machines, they weren't supercomputers either. DEC sold 400,000 Vaxes over the product line's history. Scraping together another 12ish of them at need in 1998 probably wouldn't be an insurmountable task.
That being said, these stations were designed to track lots of space and near earth objects at once. I'm not a shuttle expert, but I don't see why adding one more of those would have broken the system. In fact, if Wikipedia is right, even with one shuttle and the ISS up there simultaneously, most of the bandwidth was still reserved for US military operations*. If the US (and not so incidentally the planet it is sitting on) were about to be destroyed, I highly suspect the US military would happily donate some of it.
* - They provide a quote attributed to Space.com, but their reference link doesn't contain the quote.