@continuum I'll have to try that on the other machine (the one depending on wifi). On this one connected by ethernet cable to the router the Speedtest.net results are 658+ Mbps down and 42+ up.
I looked into downloading a new driver. The model of wireless adapter is a TP-Link Archer T4U Plus and does support Wireless AC after all. I found a new driver but it looks like it's using a much more recent Realtek one.
But after all is said and done I feel totally silly, because it turns out the stupid thing had at some point connected automatically to the 2.4 GHz band and not to the 5 GHz one. I corrected that and configured it so it wouldn't connect automatically to the 2.4 GHz band.