Why is there a Golden Gate Bridge and a road in Star Trek: Picard in San Francisco? What travels on this road? Shuttle craft with wheels?
You have asked three questions, I will go through them one by one.
The Golden Gate Bridge is still there, there is no need to tear it down with no reason
We have seen the Golden Gate Bridge several times in Star Trek. I can't remember if we saw it in the future during "The Voyage Home", but we can see it clearly in Discovery's episode "Such Sweet Sorrow":
You can see that it's covered with solar panels. Note that we don't know if these solar panels are still in use, they could be historical remnants the same as the bridge itself. There is no reason why this bridge, even if it's unused, should be torn down - it's an iconic landmark and will still serve as a monument even if it's no longer used.
Nothing travels on these roads specifically
As you can see, nothing travels on that bridge because it's covered with solar panels. We don't know if they are still in use or if there was a point in the past where having them made sense (before the discovery of nearly unlimited energy sources such as matter/antimatter reactors), but they are there.
There are roads because cars are still a thing.
Cars are still a thing in the future. And why wouldn't they? It's just not feasible to take a shuttle everywhere, and there are people who are afraid of beaming. Also, people might still be driving cars because they enjoy them.
Even though it's in an alternative timeline, we can see that Kirk drives a car, that there is something similar to a motorcycle and that there are even still hitchhikers. This all pops up in this video:
As you can see from VOY: Endgame, the Golden Gate Bridge of 2394 houses some manner of travel system, comprising individual small cars traveling at high speeds inside a series of transparent tubes, possibly some kind of mass-transit facility.
Trains on the bridge
I can't cite canon for this, but you can plainly see the "urban density" of the Marin Headlands and Sausalito, and you can bet if those are that built-out, the Presidio is denser still. This creates a transportation choke-point.
The devil is in the deconfliction. Yes, air is 3-D, but you need a fair amount of space to deconflict traffic to such a focused destination. Simply because wind conditions could hurl a vehicle off course, aside from other failures like those who like to hand-fly. Transporters, well how much "airwaves" does a transporter take? Can you have unlimited transporters in a high-density space? I doubt it.
We've seen the pods traveling on the bridge's upper deck in the Voyager scene. However such pods simply lack the density required to bring in hundreds of thousands of workers in the Starfleet headquarters and Academy complexes. (To say nothing of ordinary commuters merely passing through between non-Starfleet activities in the City proper and their homes in Marin, Sonoma and Mendocino counties). A train that handles 1000 people every 4 minutes, that's the kind of thing you need. And that's an ordinary subway train today.
So we can safely infer something like that exists, either tunneled (though the Golden Gate channel is fairly deep, making that hard) or on the lower deck of the Golden Gate, as per the original vision for the BART system.
It may, in fact, be precisely BART. The trouble with changing out a tunneled rapid transit system to a totally new technology like pneumatic tube or maglev, is that transit system is there for a reason, and the reason doesn't take a holiday for a couple of years while everything is rebuilt, debugged, torture tested and shaken down. The community simply cannot bear the cost of taking the system down that long. It isn't modernized because it can't be. So those 24th century commuter trains may be using grav-mag grapples to float above standard old 5'6" broad gauge steel railroad track, but standard they may still be. The upside is the Historic Fleet gets to operate from time to time.