Parliament has proved itself incapable of delivering what the majority want, so desperate measures to knock some sense into them is needed.
Technical it's the majority of those who voted at that time and only on the question put to them.
Since then people could have changed their minds (that goes either way, so it could lead to a larger majority for leave or a change to want to remain).
However even that doesn't answer the question what do the majority want in terms of deal or no deal, and like the TV program we've had a large figure out in front of us as to what we may get. However no one knows what will be the outcome will actually be. The closer to the end the easier it is to see what we could be left with, but no certainty until it is over.
As to whether or not we should have a republic, the problem with bringing one in because of a crisis is that the written Constitution would likely have elements designed to stop such a crisis from happening again. However this could have far reaching consequences, for instance the USA's rules about gun ownership is based on there being no standing army (so that the president couldn't have the control which the British King had displayed). However it now causes problems with regards to what if any controls there should/shouldn't be.
Although the Queen doesn't use her power; she is still a useful instrument with regards to checks and balances in that if a PM lost it and started doing crazy things, for instance becoming like a dictator, there could be ways in which she could act.
You also have to bear in mind what will happen to the Crown Estate. In that currently 25% of the income from it goes to funding the royal household. Now a fair chunk of that would still be needed for the upkeep of the Royal Palaces.
Assuming that the royal family were asked to move out (and they have their own private property which they could live in) a lot of those costs would still be needed.
Then you've got to consider the work which the President would then do, and where their official residence would be, with all the costs associated with that.
As such it's unlikely that the costs would be much different.