Resonance is a problem too. Would be embarrassing if your fancy Skyscraper Bridge fell down because of that.
Resonance shouldn't actually be too much of an issue, and it's arguably less of an issue for a bridge like this than a conventional bridge. The go-to resonance disaster is the Tacoma Narrows bridge, but it's important to note that resonance (likely) wasn't the only real cause there, as the elasticity of the wind amplified the bridge's oscilaltions (like pushing on a swing). A lot of research was actually done in that field in the immediate aftermath. Such a thing
could happen to a Skyscraper Bridge's span, depending on how it's built. If you build in passages that allow air to pass through or use a truss-based deck, so the wind isn't forced to go solely above or beneath, you'll almost certainly sidestep the problem. However, that might reduce how high you can have buildings on the span.
The artists impressions do seem to favour a more solid design, as opposed to a truss, though. Installing strakes on the span on the bridge might alleviate the wind problem, though, as is common for large chimneys and the like, just horizontal instead. It would hamper the Art Deco design somewhat, though.
But a "skyscraper bridge" that has buildings on its span is going to be much wider and heavier, and therefore should be less likely to suffer the same fate. Your two main culprits for resonance issues are wind and people walking in step. For something of that size, as long as you don't march the whole US Army over it in step, you should be fine, and you can always put a "Soldiers Must Break Step" warning on either side as many other bridges have.
Structurally, I think a Skyscraper Bridge would be sound, (and any issues like that would near certainly make themselves known during construction, though whether nature's warnings are heeded is another thing entirely). If nothing else, having skyscrapers for the towers should work.
Noise pollution for the residents would be a problem, though (not that that's stopped NYC in the past). It'll drive property prices right down, which isn't ideal, and limiting what kind of vehicles can use it is going to reduce its effects on congestion. That being said, if you're building a dozen of these, each reducing congestion a little bit might actually result in a decent sized net change.
But I can't really see anything like this taking off outside of NYC. Aside from the fact that the city itself is unusually well suited to having skyscrapers due to its bedrock, I don't really see the practicality for most cities. Granted, the truth of the matter for most skyscrapers is that they're good basically for prestige and little else, and yet they remain prolific. Skyscraper Bridges would be pretty darn costly to build and maintain, and I'm skeptical as to how useful they'd actually be in the long term. I doubt they'd remain more than a curiosity (at least, for ones with buildings on the span; tower buildings could take off), and would provide yet another tourist attraction for the alt-NYC. That universe probably has a Tom Scott video on them, I'm guessing.
In short, while cool and (most likely) architecturally sound, they're probably impractical in practice.