To provide a good education, public school systems need all kinds of supplies, products, books, services, etc. that don't make sense to produce in-house. This is where the private sector comes in -- though it's hard for small companies (who can often give the schools the best value for their money) to build relationships with schools, get through the red tape (getting onto vendor lists, going through bid systems, etc.), so often the schools are getting lower-quality supplies/services for more money.

...so wouldn't it be nice if there were some group that would help new education-focused companies to get their innovations into the (free) public schools?

I hope this makes more sense now.