I would contact some mothers with kids in the same organizations as my older one. Then pay them to let her ride home with them after activities to hang out a while. Is she close enough she can walk home? If so she might just go home.
At age 10 a kiddo should be learning home alone time and many kids her age are latch key kids. The Red Cross teaches a babysitter's class for 12 year old kids to learn how to babysit the right way.
I always figured that if a kid was supposed to be old enough to babysit someone else they should be proficient at watching themselves first and have much experience at it.
When our 11 year old is home every neighbor that I trust knows it. This way they all keep an eye out for strangers or if she's outside doing something not allowed, like being outdoors or having the front door open or answering the door while home alone. She is not allowed to turn on burners but is allowed to cook in the microwave and she often uses the microwave while we are home. So we know she knows how to properly use it. Especially since she set the last one on fire by trying to cook frozen bread sticks on an aluminium pan in the microwave.
In a year she will have enough experience to know how to watch someone else while also watching herself because we've allowed her that possibility.
If you don't live close to the school at all perhaps you can have another parent whose child is in the same activities drop her off at home afterwards.
I think at her age she should be able to work towards this.
As for your son, he needs a full time care giver. If you can't arrange his classes for a time when you are home then maybe he shouldn't do them? I think finding a babysitter who has transportation is not a bad idea all in all.
You are hiring a baby sitter, not a nanny. A nanny is someone who comes to your home and does many things, not just pick up the kids and tote them places.
A part time nanny or babysitter do not get a salary or benefits or paid more than minimum wage. They are temporary contract people that you hire for a service then you're done with them.