Assuming the person using the camera has taken basic photography classes will the Canon EOS M50 Mark II take better pictures of children outside then an iPhone 13 Pro?

That's not really enough information to go by. But while an M50ii might be more responsive than an iPhone 13 Pro, it may not be ideal in a lot of ways. It's more expensive, it may require a lens to more easily shoot this subject that it doesn't come with. And it may still depend on the shooter's reflexes and ability to draw a bead on a moving subject and operate the autofocus system to get things in focus.

In addition to all this, popular opinion and rumor in 2022 is betting that the EOS M line of cameras may be discontinued in the near future if Canon builds out the EOS R system to have entry-level $500 bodies in the lineup. EOS M's EF-M lenses, unlike Canon dSLR EF/EF-S lenses, cannot be adapted for user on the EOS R mount.

Assuming she takes a basic photography class will this camera take noticeably better pictures then an iPhone 13 Pro?

It depends on what type of pictures we're talking about. There are reasons that smartphone cameras have taken over in the snapshot department and that camera manufacturers no longer develop many small-sensor low-cost point and shoot cameras. What the higher-end cameras give you are bigger sensors and more controls. But using them to best advantage is still up to the photographer. You're kind of asking a question like: "is a violin better than a recorder? Which makes better music, assuming you've taken a basic music class?"

Just as a violin can make nicer music than a recorder, an interchangeable lens camera can make nicer photos than an iPhone. But it's also harder to learn, will require more time to master, and in and of itself isn't automatically better. It won't backup to the cloud or fit in in a pocket, and it won't always be with you the way a phone is. But the phone camera can't change aperture settings, is much more limited on lenses, has a much smaller sensor, and relies on computational processing modes to accomplish what happens physically with an interchangeable lens camera.

I'd like to get her an entry level camera with the understanding that if it's something she uses regularly we may decide to get a more expensive one in the future. I think she may be underestimating the convenience of a smart phone so don't want to spend $2000+ on a fancy camera with multiple lenses if a non-smartphone camera is not something she will use heavily.

Just me, but you're not avoiding the $2k "fancy camera with multiple lenses" issue if you get her an M50 Mark II. That is a fancy camera that can use multiple lenses that may easily cost $2k to build a basic system with. Any new interchangeable lens camera pretty much is. Even the $500 ones.