In order to maintain its influence in the world, Britain needs more than just military and political power. The Royal Family is a valuable tool of appealing to the international stage in a way that transcends the divisive nature of politics.
Britain is synonymous with its royal family. This goodwill that the Royal Family generate towards the UK is hard to measure, but having the Queen as an apolitical figurehead of the country is better for public perception than a divisive, scandal-ridden Prime Minister. Furthermore, the benefits of having apolitical and well-regarded diplomats is a general improvement in relations between Britain and other nations. If Prince William can make a good impression on Donald Tusk, or on the people of Brussels, perhaps the Prime Minister won't have as hard a job negotiating the Brexit deal. Having this form of soft power is deeply important for the UK. Losing the Royal Family will deprive the UK of a rich vein of cultural capital it otherwise profitably mines.
The Royal Family is a poor way of maintaining influence in the world. We have diplomats and public figures who can do the same job at a fraction of the cost. The image royals have generated in recent years has been poor at best. Members of the Royal Family have consistently been a source of embarrassment and international faux-pas - from Prince Phillip's offhand racist remarks on trips abroad to the scandals involving Prince Harry's nude bum. The Royals are not trained, qualified or accountable for the scandals they might create and are not the agents we want representing us.
[P1] Maintaining political influence on the global stage should be a valid goal for the UK. [P2] Soft power is a crucial way of maintaining British political influence, and we shouldn't undermine it. [P3] Abolishing the Royal Family would severely undermine British soft power. [P4] We ought not to abolish the Royal Family.
Rejecting the premises
[Rejecting P3] The Royals are not the only agents of British soft power. They are ineffective and replaceable.