“Eh. Based on the context, I would say that’s a weak proof.”
OK, you can do that based on the context, but based on the fact that that’s how every major Rabbi has learned it, I would say it’s a pretty good proof.
There is a kabbalistic inyan to have 2 head coverings. Not a halachah, but no reason not to fulfill it since it doesn’t interfere with any halachah. I do not know the source for this, and I will not be looking for it because this was NOT supposed to be a halachah thread!