The issue is, from my experience and research, that you cannot sign up just to be a respite carer, you need to meet all the same criteria as if you were signing up to do it full time (one adult home almost all the time and not working, proper spare room set up as such), which automatically rules out loads of people who would actually be a great edition as a wider member of a foster family (as I've described), providing day trips, occasional overnights, babysitting etc.
The way respire currently works in lots of places, is there's no consistency due to it being in such high demand, it means for lots of foster kids they feel like they're being sent away as FCs need a break, not that it's something nice for them. The kids don't get to know the people providing the respite, and they don't get to know the kids particularly well either. It also means when FCs eventually get respite they're absolutely desperate for it, and can end up using it to recover from being exhausted, rather than as the break they intended.
By a scheme, I mean something as I've described with new respite carers being paired up with families (either on the basis of the FCs or the kids), and supporting the same family on an ongoing long term basis, providing a set minimum amount of respite overnight breaks and giving other support too.
From what you've described above, it's the exact situation this type of scheme could avoid, you could have rang your assigned 'Foster Aunty', arranged a convenient date for your foster child to spend a couple of days with her, and she could have let the social worker know the dates.
I think this kind of thing would actually bring in a lot of new respite carers, if it were promoted properly, particularly those who've comment things above like 'we'd love to, but don't have a spare room' or 'we'd love too, but cannot commit that much time', which would probably stop full time FCs getting burnt out.