If there's not enough parking, let the market fix that problem - if private parking lots are profitable use of land, then people will build them. But people will never be satisfied with private parking lots if they could have used free public parking instead (if they'd just looked harder for a parking spot).
In other words, giving out free shit discourages people from buying that shit. Which really shouldn't surprise anyone.
Cars aren't cheap. Reducing the overall space devoted to them and utilizing that space for useful purposes would be a net benefit. Providing public transport services with that increased revenue would leave the vast majority better off financially, not to mention the quality of life improvements.
I am fine with parking lots as long as they're not publicly funded and are taxed just as every other parcel in the city. And on-street parking should be prohibited.
Congestion pricing is already a (crude, but still rather effective) way of solving that problem. A more perfect (though impractical to enforce) solution would be charging by the meter-second of lane space consumed, with rates varying over time and location depending on congestion.
Under that pricing scheme, if driving in circles were still somehow cheaper than parking then cars could drive in circles to their heart's content without causing any issues.
No, there's definitely a better solution for this - using self-driving trains to move people.
Don't get me wrong, I'm a huge fan of reducing car dependency, I'm just not optimistic that self driving cars will help.