As you say later in your post, that does not apply to non-EU nationals resident in the UK or Ireland, nor does the right to travel within the Schengen area apply to the UK or Ireland, since these two countries are not in the Schengen area and don't issue Schengen visas. (There are other EU countries not in the Schengen area: Cyprus, Romania, and Bulgaria; and there are non-EU countries in the Schengen area: Norway, Iceland, and Switzerland; and there are some "funnies" as well.)

I only post this here in case someone reading this thread gets confused: it's not relevant to the OP's situation at all.