Socialism just advocates for collective ownership, without specifying the scale or scope of that collective. It does not have to be the state government, and could, for example, be a group of football fans.

Public school systems construct stadiums and fund football teams.

Private universities also construct stadiums and fund football(and other sport) team.

Why should a sport team has anything to do with an educational facility? How does a football team...say...further the education of its college students?

How about the interstate highway system then? As far as I can tell, it is pure socialism.

The interstate system is a tax and spend program. If you're equating government spending with socialism, then you should probably revisit your understanding of each.

Some classify roads as a public good[1]. Of course, there are private roads as well.

The government owns the interstates, but that is just property, not a means of production.

[1] https://en.wikipedia.org/wiki/Public_good

It is also in some way, a disaster, but I think that's more about implementation rather than the use of public money.