Ireland is a complication to the whole thing, I'd agree. However Ireland does benefit more from freedom of physical movement (i.e. the CTA) with the bit of the UK it is physically joined to.
It's worth noting for the avoidance doubt that "freedom of movement" can be quite a semantically-tortured phrase, and it's important to draw the distinction as in the original post between "freedom of physical movement" (that is, the absence of routine controls) and "freedom of movement" in the EU sense (that is, the right to pass through controls). There is nothing at all preventing the co-existence of routine controls with freedom of movement - as of course that was the arrangement up until the time the UK left the EU.I'm with you that if the land border is removed, then the issue is massively simplified, but that's not where we are now.
I also agree that Ireland's priority will be maintaining free movement with the UK, but I would also contend that Ireland has shown it would prefer to be in Schengen.
Technically the CTA functions in the same way; it guarantees the rights of British and Irish citizens to freedom of movement throughout the CTA, but conveys no explicit freedom from controls. Indeed the Irish government's official position is that crossing from Northern Ireland into the Republic without controls is only available to British, Irish, or EU/EEA citizens - persons of other nationalities are (on paper, at least) required to present themselves to the Garda or Irish Immigration Service after they've crossed.