There have been private streets and alleys up for sale like this before in San Francisco.

  https://sf.curbed.com/2015/10/14/9911570/unicorn-alert-san-francisco-driveway-for-sale-asks-50k

Monetizing these things can be very difficult because the adjacent property owners often have very strong arguments for an easement.

The laws of easements and adverse possession are highly fact based so it's not worth opining on this case without more facts. But the takeaway is that you _always_ want to have an explicit agreement with people who are using your land, and ideally you want them to pay an annual rent that you enforce, even if it's a token sum of $1.[1] Once you allow people to use your property without an explicit agreement, you open the door to either outright losing your property or losing the ability to control your property.

This is a good thing because, on balance, if somebody willingly sits on their rights for a long period of time it's better that he lose out rather than permit him to, out of the blue, disrupt the settled expectations of everybody else. But it does mean that as a property owner you have to remain at least minimally vigilant.

That's why these streets sell for such little money. If the ability to monetize ownership of these private streets were clear cut, you can bet that they would sell for much more at auction. I doubt this couple were the only speculators who saw this opportunity.

[1] Don't think of paying a token sum as a pointless ritual. In the eyes the law, the regular exchange is objective evidence of the existence of the contract and its continuation.

Totally arm-chairing this here....

Adverse possession seems to require an element of you know, possession. No one possessed this property except the HOA until the nice couple bought it, and then their ownership has been the opposite of notorious because they didn't want to come into the light until they had their legal ducks in a row. If the couple had been paying the taxes for the last 30 years it might come under adverse possession. If this goes in favor of the couple without limit to whatever they want to do, it would be because the auction is unreservedly upheld. I basically agree that there is some chance that at least an easement will be found to exist in favor of the HOA. The tax collector did sit on their rights to the detriment of all, including a bit of degradation of the already weak reputation of tax collectors.

  My question is, why didn't the city just convert it into a
  public road?

Because then they'd have to pay to maintain it. The city would be exchanging revenue for a huge liability. Plus, that'd probably be the worst outcome for the homeowners because then the road would have to be kept open to the public.