What exactly would be the problem with a small amount of ferrous material inside a QFH GPS antenna?
I wouldn't worry about ferrous metals near the antenna. As you suspect, it's probably a bit of superstition carried over from low frequency loop antennas, where it's more important.

First, the QHA is a balanced antenna, so there shouldn't be any significant currents flowing on the feed line in the centre

Then, it's not as high Q antenna, so won't suffer unduly from a more lossy material. In fact a GPS antenna won't notice a fair amount of loss - the typical little square patch antennas are less than 50% efficient, and the linear polarised chip antennas used in cellphones, for example, are even worse.

Finally, steel and stainless steel get a bad rap in antennas. They're not good for making very short or small transmitting antennas, with high Q and large circulating currents, but on a resonant dipole, even at 7 MHz, the loss isn't too bad, considering the strength, cost and corrosion improvement over copper. At higher frequencies, a lossy material in the antenna matters even less.

You should worry, though, how the QHA is fed - it needs a balanced feed. There may be a sleeve balun lurking in the brass pipe. I haven't seen the video. If it's just fed directly from the coax, the patterns won't be ideal.