Replace jockey on old XT RD-M750 or change out to RD-572?

I presume you are referring to the jockey wheels on the RD. These are not called 'sprockets'. They are not ceramic, though the guide wheel has ceramic bushings, while the tension wheel uses sealed bearings

The M750 RD is all aluminium and in comparison M592 is part steel (so slightly heavier), and uses cheaper jockey wheels. However it sounds like your jockey wheels are used up.

The other thing is M592 is Shadow which means no top pivot, which is a common wear point. Both are good, probably the shadow design is better, whereas M750 is a better implementation of the non-shadow design.

A new Deore RD will certainly be much smoother than a worn out XT RD. XT is more durable and lighter, but a worn out part is a worn out part.

If the RD has not had much wear in the pivots, but the jockey wheel has merely perished due to age then, you could just replace the jockey wheels in the old RD. But you don't say, so I'm assuming it's very well used.

The best possible solution would probably be a new 9-speed Deore XT Shadow RD, but I'd be happy with any Shimano 9-speed RD personally. (don't buy a fake one!) If you did get the M592, then you could upgrade the jockey wheels to XT grade ones for higher durability/slightly lower friction. The few extra grams from the steel cage is not really a big deal.

My thoughts are if your keep an old cassette (and presumably chain) then the difference in performance between the old XT and new derailleur will likely be less than expected. Obviously this depends on how worn the derailleur is, but I am presuming it is reasonable condition except the jockey wheel. Exactly what to do will depend on your budget, the state of the current drive train and you/your childs patience with sloppy shifting performance

Based on this, I think you should look at two approaches and refine from there.

1. Replace the jockey wheel*.
2. Replace the lot - derailleur, cassette, cable and chain and maybe shifter.

*(Presuming you can get one significantly cheaper than a new deraileur)

In terms of gears and 1x - you say a child, is 28/34 going to give a low enough gear? If not, replacing the lot makes better sense. Consider going to a large cassette to give a bigger range. Adding a shifter means you could increase the number of gears. Microshift do a cost effective range of drivetrains of similar quality to SLX at good prices in 9 (and 10 speed). Review of Microshift Advent