Why was Monarchy abandoned by everyone who got rich enough?
Short answer: Because monarchy sucked.
The formalist position comes down to: Democracy sucks.
Of course, by "democracy", they mean a very specific, not very democratic approach as handled by most Western countries.
The simple retort is: you don't have (nor do I) a very good picture of how good or bad it was, nor do you have any reasonable measuring stick. Because the crushing poverty of the pre-modern age makes all other measuring sticks almost worthless.
You do have a picture that says that modern monarchies mostly to entirely suck as places one would like to live (I notice none of the formalists have moved to Saudi Arabia)...and that all modern places where you'd like to live have instituted partial democracy.
Very simply: you're not (nor am I) smart enough to identify the inter-relations. Nor which is better. Why do all rich countries go democratic? I don't know. Simple, Moldbuggian, monocausal theories are nearly universally wrong. Something about being rich for multiple generations makes for democracy? Something from Mancur Olson?
If you want a good system, there are lots of thoughts about what one might do. There's lots of evidence from the modern world. And there's lots of theory on the topic from many directions. Go back to something widely found to be crappy back then, and widely found to be crappy now...doesn't seem like a smart direction.
Switzerland: Tiny, almost entirely autonomous federated states, with very little power to the federal government? Good Model.
British Hong Kong and Singapore: Please don't make any laws, because otherwise we British folks might have to do something, and we'd like to ignore the place.
Post-British Singapore: Technocracy with the major electoral check of moderate democracy on continuing wealth generation for the citizenry, because if they're not happy, they can vote your ass out.
Scandanavia: WE are a cohesive social unit who has some concern for one another. Free the markets to be rich, and then share the wealth. It's not obvious that wealth-sharing is anywhere near as wealth-destructive as is, say, tariffs, regulations around starting a business, etc. Indeed, there's some evidence that for general well-being one should choose redistribution over regulation.
BBdM-ism: Centralized power has only one sane way to behave...and the game theory is painfully clear: Screw the citizenry as much as you possibly can, and redistribute loot to those folks whose support you need to stay in power. Of course, all systems are power coalitions, whether it be a democracy or a theoretical dictatorship like Saudi Arabia...so the notion of coalition-free power is ROFLMAO.
Olson-ism: All static systems necessarily trend towards sucking. Good ~= Wealth; Wealth ~= Growth; Growth ~= Innovation; Innovation ~= only partially stable system. Key to a good life/future is to ensure that systems are not very stable ~= cannot Ossify.
Romerian/LaTNB experimentalism: We so obviously (updated for missing negative) do not know what works...let's have a lot of folks do cities as startups, and see what works, because it's so obvious we don't know much of anything, given such limited knowledge. Exit is important.
Aretae-ism: As it's true in all other economic systems, scarcity is the whole problem. The only problem that needs solved is the scarcity of countries. If it were easy to switch countries...the countries would get a lot better really fast...more or less like swiss cantons did.