Both of these kids are desperate for a social life, and their mother won’t let either of them have one.

Having a second kid doesn’t guarantee that your oldest will have a friend. A lot of parents have their 2nd child because they don’t want their 1st child to be a lonely only child with no friends. You can debate over whether that’s a good enough reason to have a kid until the cows come home, but that’s neither here nor there. 

What is here and there is that siblings often do not get along, and it’s not common for them to be best friends, especially when they have a considerable age gap. A 10-year-old and a 12-year-old will be able to hang out with each other and do similar things for most of their childhood, but a 10-year-old and a 15-year-old have a much smaller window of opportunity to be playmates, and that’s okay! You really can’t expect a much older teenager to want to spend their precious free time with a little kid. They want to have friends their own age, and they should be allowed to have that!