"....so why should abolishing the monarchy cause disruption?"
Gosh, where to start?
The British monarch is also head of state in 15 other countries. Would those countries need to become republics too? What if they didn't want to? If we expect them to hold referendums to decide, who will pay? (They're expensive). What if they refused and insisted on maintaining the current model?
The British monarch is head of the Church of England. Who becomes the new head, if there is no monarch?
What do we replace the monarchy with, an elected president, or some other model? Who decides what the model should be, including the rules for who is allowed to stand? This is what happened in the most recent Australian vote on whether to become a republic in 1999. The republican vote was split between two different versions of what to replace the monarchy with, so the winning vote was to keep the monarchy.
What happens to all the agencies that are run 'in the name' of the monarch, i.e. HMRC, HM Treasury, the courts service to name but three. Would all the paperwork for existing court cases need to be changed (it could no longer be 'Regina vs')? What would it be changed to?
Money - would we need to re-issue new notes/coins every time the new head of state/president is elected?
Would Royal Mail be allowed to keep the name? What happens to stamps which currently feature the monarch?
Will businesses that have been awarded royal warrants be allowed to keep them?
This list doesn't even begin to scratch the surface of the implications and 'disruption' involved.