Depends how rich you are!
Ideally it'd be nice to see rads of every horse & if you can afford it &/or you or your trimmer are really concerned about anything, then yes, go for it. But as it's not cheap, it's only those in a bad way that tend to get them, if that. Could be wrong, but from memory & as much as can be told from an internet forum, wouldn't class your boy as that bad at all.
Yes, your gut feeling is right, that 2 months is nothing much in the scheme of things & depending on the state of his feet, management, diet, etc, it could take a LOT longer, or even never, for his feet to become strong enough to cope with rocky ground bare. If you do xray & find that his soles are paper thin or such, what are you going to do differently with that knowledge? If he's truly comfortable on more yielding ground I think that's an indication his feet aren't likely terrible though. While I'm not convinced about the accuracy Pete Ramey claims about collateral groove depth, I think 'landmarks' such as that can indeed give you a pretty fair idea of sole thickness & such. I'd just continue to diligently protect/support his feet with boots as necessary for another fair few months at least yet, before considering it may be worth trying him bare.
While I remember they're not in fantastic shape & having only been out of shoes a couple of months, so there may well be other 'issues', I disagree with Gypsy & being 'footy' on hard going is not at all necessarily indicative of pathologies any more than a person not conditioned for it should be classed as 'unsound' if they have to hobble a bit when walking bare on gravel.