I think the fact that the stars appear to be printed on a 2D surface is realistic. They're all so far away that you won't see any parallax, but you will see time-varying distortion from gravitational lensing. Your brain would probably interpret that as a 2D surface. (The night sky on Earth looks like a sphere even without the time-varying distortion.)
I suspect the lack of visible Doppler shift is also justifiable. There can't be a huge Doppler shift because it would be potentially fatal. If the Doppler shift can be limited to a small value then it can probably be limited to a value so small that it can't be seen – especially when we're dealing with phlebotinum that can magically produce whatever spacetime geometry we want.
A traversable wormhole isn't a black hole. Black holes have lightlike geodesics that circle the hole arbitrarily many times, creating infinitely many images of background objects, but I don't think that needs to be true of a traversable wormhole.
Watching the YouTube video, I see a number of other problems, though.
At 0:25 one guy says "it's a sphere" and another guy says "well of course it is", and then explains that it's a circle in the 2D analogy with the folded piece of paper, so it's a sphere in 3D. That explanation is correct as far as it goes, but there's really no reason to expect the wormhole to be spherical, and there's a good reason to expect it not to be: spherical symmetry means that any path through the wormhole must pass through the exotic matter, and through whatever infrastructure is needed to keep it in place. It would make more sense to have a configuration where the exotic matter is kept away from the path you follow with your ship. The only reason Thorne et al assumed spherical symmetry in the paper was to simplify the math.
Relatedly, it doesn't seem very plausible that everything necessary to maintain the wormhole would be absolutely invisible (and seemingly intangible as well). Again this is just a simplifying assumption from the paper, not an expected property of real traversable wormholes.
They're shown orbiting the mouth quite rapidly, but then once they're "inside" they seem to have no significant angular momentum any more. I doubt they're following a spacetime geodesic. It's probably a non-geodesic path that the special-effects people thought would look exciting.
At 2:34 there's a sudden flash of light, as though they are at that precise moment entering the wormhole. Everything should be gradual and continuous. There is no event horizon in this sort of traversable wormhole solution, nor any other well-defined boundary between outside and inside. Even if there were a horizon, nothing would visibly happen there, but there isn't.
Also at that moment, there seems to be visible distortion of the ship itself. Distortion like that is not a harmless optical effect, it would pretty much be like taking a wrecking ball to the ship. The whole point of these wormhole solutions is they don't have any significant local distortion.
When they're inside there are lightning-like effects that are probably not motivated by any real physics.
At 2:58 one guy says "we're passing through the bulk". There's no "bulk" in the Thorne et al wormhole. The spacetime of general relativity isn't embedded in a higher-dimensional flat space. If they're talking about a brane-world model or something of that sort where there is a higher-dimensional spacetime, they still wouldn't be any more "in the bulk" than they usually are. The protons and electrons that they're made of are attached to the brane; they can't leave it. (Nor can they see the bulk, since photons are also stuck to the brane.)
The same guy says "controls don't work here". Again, the spacetime in the wormhole is the same as spacetime everywhere else; that's the point, that's what makes it safe to traverse. The instruments might be a bit confused, but they could still fire their thrusters, or whatever that guy was trying to do.
Starting at 3:16 there's a patch of optical distortion inside the ship (and somehow perfectly stationary with respect to it). One astronaut says "distorted spacetime" and I guess we're supposed to assume he's right. Another astronaut reaches out to touch the distortion; that's implausible because no one dumb enough to do that would qualify to be an astronaut in the first place. Her fingers visibly distort (by a huge amount). There's no way that whatever is bending the light that much wouldn't also snap or crush or shred her fingers.