Well I won't be watching as I have a prior engagement, but if I was watching I would.
For me the crown is symbolic, and the monarch a symbol of that. It symbolises a shared history, a shared future and represents that we are a collective and not just separate individuals. If you see it this way the monarchy is uniting.
It's nothing to do with the monarch as an individual, you'll get good and bad monarchs. They are not better or superior people, but they represent the values we should collectively hold as highest, and this is a role and a duty assigned to them.
Because that's how I see I'd have no problems swearing allegiance, I'd be swearing allegiance to the symbol and what it represents not him personally.
It's like Americans swearing allegiance to the flag. We just embody this into a person who has this symbolic role.