I'm not sure about the level of power etc held over British society - many British people would be happy to see the monarchy go. I personally don't see their relevance (I'm in the UK). The queen was maybe a special case but I don't see Charles and the rest gaining anywhere near that level of 'adoration'. When the queen passed away many people queued to pay their respects. There were also many people who rolled their eyes at the idea of spending 18hours in a queue. I'm sure many of the mourners were genuine well wishers but there were also people who queued just so that they could say "they queued!". It will be the same with the coronation. It will be reported that the British people were falling over themselves with street parties galore - in reality, lots of folks will be enjoying an extra day off work with no interest in the coronation whatsoever. (I also suspect that the reason the majority of shops chose to close on the day of the queen's funeral wasn't to give their employees the opportunity to pay their respects but to avoid any possibility (or heaven forbid, national embarrassment) that folks disinterested in a royal funeral might instead choose to do some DIY shopping. How bad would that look for a country so loving of their royal family?). If the British monarchy disappeared overnight there would be an economic impact in terms of job losses (tourist related/royal households etc), charities would need new figure heads to help boost donations, the "OK/Hello" type magazine would be short of a few glamour photos and the press would have less stories to print. Aside from the negative economic impact, I struggle see any downside to them not being there. Hopefully an upside would be a reduction in the level of hate aimed at other human beings and surely a world with less opportunity for hate is a good thing.
I can think of two instead of a none corruptible entity the UK would have to pay out for what are often very corrupt Presidents who wouldn't have the pulling power to bolster trade or time to support charity for UK citizens. The Queen now King provides a third voice possibly of reason or devils advocate for any serving PM.