Some do, some don't. Last time I bought a satnav I had a problem finding one that offered "waypoints" at a reasonable price. It is assumed that if you want waypoints you must be a delivery driver, dropping off stuff at multiple points during the day, and therefore the satnav has a "professional" price.
You only need to start thinking about that to realise how impractical it is, and it is in fact one of the main problems with SD cars. Many owners driving anywhere near city centres will not park them at or near their away destination - they will send them empty off home again, or somewhere else, and recall them when needed. That doubles that traffic at a stroke. Add to that the owners who were previously deterred from driving into cities by the parking problems (and hence using public transport - think London) will now do so - because those problems would be avoided, as you say.
Apart from that, it is difficult to find a parking place at any price even in the suburbs these days - and that is without the practice of sending empty SD cars to park there as well. I live in the sticks about 10 miles from my market town, and the only places to park there are the supermarkets (free but patrolled to catch non-shoppers), a couple of council car parks (the sort you propose to avoid, usually full anyway), the modest suburbs (full of yellow lines and/or parked cars already), the station (20 minute limit), and a few country lay-bys 3-4 miles out, with typical capacities of four cars at most.
What would happen with SD cars is that the owners will send them to look for a parking place with the result that hundreds of them, empty, will be driving in circles all day. After all - electricity for EVs is cheap/subsidised/free, isn't it?
The mass adoption of SD cars (and EVs) will need to come with a radical revision of road rules, laws, infrastructure, taxation, and a fundamental questioning of what roads are really for anyway. They will not be a simple slot-in replacement of driven cars. The change will be even more radical than that which occurred with motorised vehicles replacing the horse-drawn.