The RF spectrum is a common good, not a public good. Public goods are non-excludable and non-rivalrous. The RF spectrum is non-excludable (anyone can transmit on any frequency, given the right equipment) but rivalrous (transmitting on one frequency prevents others from using that frequency).

Requiring the winner of a spectrum auction to use it is a way to prevent anti-competitive tactics (since the government is granting a monopoly to the winner). The goal is to incentivize productive use of limited resources, not necessarily to benefit everyone. In theory, the winner could use the spectrum for entirely internal purposes. Though in real world spectrum auctions, the government usually has stipulations such as requiring interoperability or using open standards. This reduces the value that the government captures, but likely increases the value that is created overall.

Before spectrum auctions, the government simply mandated what frequency bands were used for what, and by whom. Getting access usually meant lobbying and back room deals. Sometimes the FCC used lotteries, which caused speculators to enter lotteries and then license access (basically capturing revenue that would have gone to the government had the spectrum been auctioned). In practice, auctions are the worst form of spectrum allocation, except for all the others.