Plenty of soldiers (sailors, airmen) get time off for 'other' duties. It's not peculiar to the Royals.
I have a friend who has just competed in the Commonwealth Games. For the last 6 months she has had 2 days a week 'off' for training. She finished work completely mid June to train full time, went to the games, and is not due back at work till 1 Sep.
I know another soldier that has a day off every week during the football season because he is a trainied physio for a 'proper' footy team.
For 2 years running I had 2 months off to train and then compete in the Army Snowboarding Championships, and know people that had more in the lead up to the Winter Olympics.
We have soldiers permanently training (and not doing their 'day job') for sports such as bobsleigh and skeleton.
There is even a unit, nicknamed "Sports and Pastimes" where serious athletes get posted because they can guarantee time off for training and competitions.
In my career I have had time off for expeditions and exercises such as a Reef survey in Malaysia, trekking in Nepal, diving in Malaysia, Egypt and Gibraltar, and snowboarding in Canada. I have been sent to NZ at short notice to assist with a legal issue and travelled back to warzones as a civilian to check stuff we did was still in place.
All of the above was at MOD expense.
One Royal Engagement "on duty" a week? Pah, chicken feed 