I don't think the union workers want to make their jobs harder, I think they have no reason at all to support using additional automation and technology to help do their jobs.
You haven't explained how government-run ATC is safer than private ATC, either inherently or de-facto. Did you watch the video? Canada has the second-busiest ATC system in the world and it's run by a private for-profit company. Do you think the US's ATC needs are so vastly different than those of Canada that there is nothing to be learned here?
Is your concern that the airlines would essentially capture the private firm and have it do their bidding? Require almost no dues from the airlines and push things too far such that safety is compromised? _Everything_ gets out. If there were a crash and it came to light that the airlines behaved this way they would be facing the mother of all lawsuits that would totally ruin their businesses.