I don't think this argument refutes the previous point (which I found surprising, but I think is a good one). Normally, if a competitor lowers prices to undercut me, they will sell more widgets because people will switch to them. But with price matching, they know they won't sell more widgets (because people won't switch), so they keep their price high, which enables me to do so as well.

This is of course a very stylized/idealized argument, but I think it's a good point to consider.