Parking becoming difficult and/or expensive pushes patrons to places where that's not the case.
/s
And the 'spend' from people driving in from the suburbs to go shopping will easily be overshadowed by greater economic activity resulting from a denser urban core.
That's not really how it goes. This assumes that if you make it painful for people to do what you don't like (park), they'll stay and live the way you want.
But no, people will simply abandon the area due to inconvenience. The area I mentioned above (previous post) went from a nice affluent residential area to a dilapidated area where only the poor and university students (also poor) lived. Everyone who could afford to leave, left.
Humans will always look for a better solution for themselves unless you impose a complete dictatorship. It is not possible to advance a cause (whether it be less cars or more public transit or better EV support or better bike paths, etc) by simply making the status quo inconvenient. People will simply avoid it.
You advance a cause by making the new alternative actually more attractive, then the majority of people will happily adopt it.
Good. Then more people will choose different forms of transportation or internalize their costs. If parking becomes scarce, providing parking will become more profitable and people will pay the actual (enormous) cost of temporary car storage downtown.