Obviously, finding something better is an enormous topic which a post on Mumsnet can't really scratch the surface of.
My starting point would be an apolitical elected head of state - in a non-hereditary position, the elections held perhaps every 10 years - but with a clause that a 'vote of no confidence' could be passed in certain circumstances, if they were guilty of the equivalent of 'gross misconduct'.
Their remit would focus on charitable and diplomatic activity; the connection with Parliament would be removed. Roles such as head of the COE, Armed forces etc. would pass to senior figures in those organisations.
The elected head of state would be provided with accommodation and staff suitable for diplomatic entertaining, family accommodation and very modest 'bases' elsewhere if needed; e.g. a flat in London - all occupied only for their tenure in role.
The present Royal Family would give up their state funded accommodation and this would either be fully opened up to the public on a 'stately homes' basis, or put to other purposes such as accommodation for use by charities.
Incidentally, there would be nothing to prevent William standing as elected Head of State if he wanted the position. The key point is to remove the hereditary element and give the public the ability to choose the person to stay or go.
Remember that we were only one tragedy away (until the birth of William) from having Andrew as our present king. That, for me, is one of the strongest pro-republican arguments.