I'd add to that!
I have lived and worked in four countries (three continents):
1 - the UK, unelected HoS, elected Parliament (largely - House of Lords has some influence but can be over-ridden by the House of Commons), independent judiciary
2 - elected HoS, elected Parliament (or equivalent), independent judiciary. Established democracy (well over 100 years)
3 - elected HoS, elected Parliament (or equivalent), some political input in judiciary but largely independent. Established democracy (well over 100 years).
4 - elected HoS, elected Parliament (or equivalent), independent judiciary. More recent democracy.
In every one of them people who are rich and well-connected have disproportionate influence over political decisions, including those affecting taxation, human rights, welfare benefits, state education and healthcare, or any other topic you want to nominate. AND wealth and connections continue over generations and are largely self-perpetuating with a degree of social mobility.
There was a period in country 4, during and following the establishment of democracy, where there was more movement in political and socio-economic terms, but the longer the current set up continues the less movement there is. Certain families/communities/areas have developed and consolidated their wealth and status.
I don't see any difference between monarchies and republics when you look at wealth inequality, or social mobility.
So I don't see any evidence that changing constitutional arrangements in the UK would do anything to change the problems of inequalities in the UK.
It is a wonderful smoke screen to distract people from the power they do have to get involved in the current set up, and do what is open to them to campaign for change and use the levers that do exist to effect change.
The problem is not people lobbying for loopholes that will benefit them, the problem is the government - and elected politicians - agreeing to it when it comes from their cronies and the 'inner circle'. That includes someone paying to go to a political party fundraiser so they can have quiet word with a Minister about whatever legislation is getting in the way of them becoming even richer.