Are spectrum auctions a tax?

According to Wikipedia,

A spectrum auction is a process whereby a government uses an auction system to sell the rights to transmit signals over specific bands of the electromagnetic spectrum and to assign scarce spectrum resources.

In 2000, for example, the UK government auctioned off five 3G licenses and raised a total revenue of $£22.5$ billion — or $2.5$ percent of the UK’s Gross Domestic Product at the time. In The Undercover Economist, Tim Harford argues that the auction could not have resulted in higher prices for mobile phone users, because

firms will charge the maximum they possibly can, under any circumstance [whether the licenses are cheap or not]. We also know that their ability to do this is limited by their scarcity power… In the UK there was scarcity built into the amount of radio spectrum available:
five licenses were the most the technicians could offer. It does not make any
difference to the customer how much the licenses cost, but it does matter to the
taxpayers, who would like the government to get a lot of money for this valuable
public resource.

Let’s say for the sake of simplicity that each license sold for an equal price, $£4.5$ billion (the prices varied, but this doesn’t affect my argument). This amount had to come from somewhere. Let’s say they borrowed this money. The bills they send to their customer will be affected by three things: infrastructure costs and wages, profits, and the repayment of the loan used to pay for the license. In a free market, the companies would have roughly the same costs and profits. Harford is right that neither of these is influenced by how the licenses were distributed: scarcity is baked into the amount of spectrum available, and only scarcity power determines profit.

But consider the last of these costs. Over time, companies would recoup the licenses’ cost by charging their customers. Sure, they could postpone this recoupment, but they will get it someday (and will pass on the interest to the customer). It should "make a difference to the customer how much the licenses cost". If my thinking is right, the windfall from this auction was right out of the taxpayer’s purse.

In reality, telecom companies in the UK nearly went bust after the auction. But that needn’t have happened. The telecom companies without 3G licenses were the most seriously afflicted. Elsewhere, too, telecom companies have paid exorbitant sums for spectrum licenses and come away just fine.

Tim Harford got his economics degree from Oxford. I, on the other hand, haven’t ever read an economics textbook. I’d bet the error in reasoning is on my part rather than Harford’s, but I’d like to know where.