I don't have this problem but lots of people on this forum have had different children at different schools and I think you have distilled the main point nicely "it is about finding the best school fit for each child". I have friends who have this issue because one child excels at sport or music or dance and it is quite "normal" to send them to different schools to accommodate their needs and talents. Lots of siblings attend different schools because they are single sex schools. None of the siblings in our family finished school at the same high school. So there are lots of reasons siblings don't stay in the same educational environment.
Practical issues do come in to it. How far apart are the schools if you have to do drop offs and pick ups and do they start and finish at the same time? If so, is there after school care if you need it. What about if you have to attend functions at different schools on the same days. That isn't going to happen often but your children are very young and that is when these issues crop up most. When they are older it isn't such a big deal.
As for changing schools. That one we have done, a lot. Our DD is quite reserved but she coped surprisingly well with multiple school changes across towns and states. She has friends all over the place as a result. It actually got much harder when she got older. Now, friendship issues play an increasingly dominant role.
For what it is worth, we have done public and private schools. The private schools looked impressive and offered the world but didn't necessarily deliver. We had very good and not so good experiences with public schools. It can be really hard to know until you are there how any school works for your child. Even within each school, at this age especially, so much depends on the actual individual teacher. Try and meet the person who would be teaching your child if possible and watch them interact together, even better.
