@FlorenceandZebedee
Ok, so I’m new to all of this and time and experience may change my views but I’m surprised that someone would view a request to be met as ‘indulging’ a mum. In everyday life I wouldn’t let my children have a play date with someone I hadn’t at least met/spoken to/exchanged numbers with so why would I not want to know who will be looking after my children-potentially at some points by themselves if circumstances arise which dictate this? I would always make the request politely as demands will never go down well but I wouldn’t see it as someone doing me a favour I’d see it as normal for someone who was inviting children into their lives and home to want the best for them and that would mean as good communication as possible….
Well, your tone here suggests it's more of a demand/expectation than you think it is.
First of all, why would you be surprised or offended by the word "indulge"? It just means to give you what you want, which is what would be happening here.
Secondly, childcare providers are offering you a paid for service, and as such have an obligation to satisfy you. That is not true of random people that happen to be around your kids.
Thirdly, the "vetting" has already been done by the child's other parent. I may not allow my child to go on a play date with someone I hadn't met but I obviously WOULD allow my DP, her dad, to make the call on whether they were alright to introduce her to when he was there. The fact that you don't get to be there when that judgment is made now, is a consequence of you splitting up, and not anyone else's obligation to make more comfortable for you, especially at the expense of their own comfort.
Which brings me on to my last point. Of course she would be doing you a favour, and you need to come to terms with that. Believe it or not unrelated people's lives rarely revolve around satisfying people that they are fit to be in the presence of their children, however natural you see that desire as. Due to the position you are in, you are asking for and expecting a favour.