Must be another difference between the US and UK. It seems many look at it as a chore or KMA moment for watching a child relative.
I watched my DGC for any reason. Any time. It was covered. I made sure I was there for DGC meals, loved giving her a bath, loved reading, crawling around, playing "Go this way", singing DGC to sleep.
When DD and her NVDH split, I got to be with DGD during the day, or take her home with us for night or even two. As she got older, I was there every a.m. bright and early to help her get ready and take her to school. Then the reverse in the p.m. I was the luckiest person on earth.
It was always a privilege and never a chore. I was blessed beyond measure. And the time is fleeting. Blink and they're grown.
Every single one of my friends did the same as often and as much as they could. Some still do since they have multiple young DGC.
So no, I don't "get" the relative at all and see no reason why OP should do what said relative dictates as to what times and when. OP--Hire a sitter, a lot more stable and you can go to sport class every week, if you choose, without worry. Win-win.