Yes to all of this. If they took a beach holiday on their own dime and paid for all of their own accommodations, travel, etc. then good for them and I hope they had a fabulous time. However, if they somehow didn't pay for those things themselves and instead somehow billed them as a "working" or "research" trip, etc. then I agree that the taxpayer has a right to know what they were doing.
I do, however, see and agree that it comes across as hypocritical and very "do as I say and not as I do" if they did indeed get to that privately funded holiday on a private plane. Even on private and privately funded time the BRF are still very much public personalities and while we shouldn't expect to see pictures of their privately funded holidays, they shouldn't expect to be immune from criticism and accusations of hypocrisy if they very publicly preach about conservationism, saving the plant, reducing carbon footprint, etc. and then hop on a private plane when commercial travel is readily available. Sidenote: this only applies if their security teams have not determined that there is any credible threat to them traveling on commercial transportation. If there have been credible threats or reasons to believe that they are in danger then this no longer applies. Some will argue that there's always a threat and I suppose that in some ways that's true but I think there's a difference between a security-determined and highly credible threat vs. the general "this person is famous and I'm crazy" type of threat that happens every single time a famous person, be they royal or just famous, is in public.