Elementary School - Private or state? Does it matter at this level where you send your kid?

It depends on the community.

If the public schools are overcrowded and poorly funded, then maybe a compelling case can be made for an alternative. However, another approach to this problem is to vote with your feet by moving to a different community with better public schools (even if this means a longer commute for Mom and/or Dad).

And sometimes people choose private schools for a short period of time only.

In our neighborhood, there was a period of about 5 years when there was massive overcrowding at our middle school (the kind that required as many as 14 portable classrooms) because the community was waiting for a new middle school to be built. Both of my kids went through middle school during this period. We didn’t consider alternatives because the kids had no unusual needs, and we felt they could cope with the situation. But there were other families who sent their kids to private schools for middle school only (even though they had gone to public elementary schools and would later go to a public high school) because they felt that their kids wouldn’t do well in the large classes and impersonal atmosphere of a massively overcrowded school.