I don't think the wealth disparity is inherent to monarchy as a system, no:
a) Norway, Denmark, and the Netherlands are all constituional monarchies, like the UK. They have amongst the lowest wealth inequality in the world (I think the Netherlands might be the absolute lowest, I could be wrong on that)
b) some systems of government actively promote the election of the super-wealthy, which the UK system doesn't. E.g. you'll never see a poor, or even a moderately rich person, elected as US president. It requires a huge amount of private wealth as well as wealthy donors to stand any chance of being either the Democratic or Republican nominee.
c) related to a): Many countries, including those listed above, have a pared-down monarchy where all but the immediate family are expected to go out and get jobs, sometimes alongside royal duties. They don't have a huge royal family supported from public funds.
Obviously any system which allows you to inherit wealth and property inevitably leads to a small elite of the super-rich, and almost as inevitably to at least some accompanying ideas about class, particularly when it's been going for centuries, but to change that would require more than just moving from a monarchy to a presidential system whilst leaving inheritance laws intact.
Anyway, I don't see any of this as theological, except on the level of the individual conscience. Certainly, there is a place for Christian thought in the public policy debate on taxes and wealth, but I see that as separate from monarchy per se.