I think it's actually a pretty reasonable description? Sounds like it's a paradox because we assume a rigid body, but nothing in real life is a perfect rigid body. So the situation simply becomes a bunch of particles following circular orbits. If you measure distance along one direction (while constantly accelerating in relativistic speed) you get one number; if you measure distance in another direction you get a different number. But that's what relativity does.
In other words, it's similar to the simpler question: "If I have a perfectly rigid rod that can reach the moon, and I push it, then the other end pushes the moon immediately. But speed of information cannot exceed speed of light. How come?" Answer: There's no perfectly rigid rod.
The example you give and its answer follows a line of reasoning that leads you to an interesting conclusion, that's the point of thought experiments.
What I was saying is more akin to answering your example with:
"Oh no you can't! There's not enough steel on Earth to build such thing and even if you had it, it would require an EEeEeeenOOOOrrrMMMoooUUUusss amount of energy to put in place ;)."
That would be quite a moronic interpretation of the problem that completely misses the goal of said thought experiment, which is, well, to make you think.
The difference there is, I think, that the pragmatic argument of "not enough steel" doesn't prove the non-existence; while the argument about "there are no truly rigid bodies" does.
I haven't seen that Veritasium video, but it sounds like it makes the exact same point. It's not that we haven't found a material that's rigid enough; it's that a rigid disk is counter-factual to begin with, even in non-relativistic conditions.
It's one of the reason I dislike physical terms that has words like "ideal" or "perfect" as their part. They subliminally suggest that their properties and behaviours are the "true" ones while the real material things are their imperfect counterparts whose imperfections you can sometime disregard.
Of course, it's exactly the opposite: it's those "ideal" concepts are imperfect approximations of the real things, omitting lots of details which sometimes are not that important but sometimes are absolutely crucial.
My personal favourite example is attaching a perfect source of voltage (zero internal resistance) to a perfect wire (zero resistance). You can't arrive to this scenario starting with the real world entities: both the battery and the wire will have non-zero resistances and depending on their proportions, you end up with approximating either one of those as zero, or none, but never both.
I think it's actually a pretty reasonable description? Sounds like it's a paradox because we assume a rigid body, but nothing in real life is a perfect rigid body. So the situation simply becomes a bunch of particles following circular orbits. If you measure distance along one direction (while constantly accelerating in relativistic speed) you get one number; if you measure distance in another direction you get a different number. But that's what relativity does.
In other words, it's similar to the simpler question: "If I have a perfectly rigid rod that can reach the moon, and I push it, then the other end pushes the moon immediately. But speed of information cannot exceed speed of light. How come?" Answer: There's no perfectly rigid rod.