Have played all of those but Nairn. Generally speaking, I would agree that if you’re gonna play them once ever, you might as well spring for the caddie to enhance the experience. That said, if you’re trying to keep things cost effective…
St Andrews Old Course - Absolutely need a caddie the first time around. One, because it’s the Old Course. Two, because there are quite a few blind shots and it’s relatively non-descript, so you can get flat out lost out there. Highly recommend caddie.
Carnoustie - relatively right there in front of you. If you have a rangefinder/GPS for the holes that have burns/streams dissecting the fairway, you’ll be good. Did have an ex-Walker Cupper caddie for me there which was pretty cool though.
Kingsbarns - right there in front of you and lots of elevated tee boxes. No caddie needed.
Royal Dornoch - much like St. Andrews, I’d recommend it simply because it adds to the experience of playing a world class course. That one is also fairly straightforward though, so not necessary. The caddies there are generally low handicappers, so they’re good