Your point of the extent of deformation the floor can have for the theorem to hold is valid though.
He claimed that in the early days there was a lecturer or professor there that, at least in Eindhoven, was very important to his field of expertise. If I understood him correctly, this prof's ideas about engineering mechanical systems revolved around restricting the degrees of freedom as much as possible. A three legged table cannot wobble, but a four legged table can and usually does because it is overdetermined. In mechanical systems (for instance sensitive optical mechanics) reducing "wobble" is key. And the best way to reduce wobble is to make sure it cannot occur.
Here it gets interesting. My source claimed that this professor had laid down his ideas in a standard work in Dutch, which was never translated in another language, restricting its influence to Dutch mechanical engineers. He also claimed it is not a coincidence that Philips and later ASML took an early lead in designing optical systems.
Not sure if it is true, but an interesting story nonetheless.
This is how I was taught mechanical engineering (France, 2000-2005) and not a Dutch in sight.
The table does not rotate around its own axis, but rather it rotates in "such a way that three legs stay on the surface", i.e. moves around in 3d with a surface-contact constraint, which seems like a motion with more than one degree of freedom to me. Is such a rotation always possible? Is the motion somehow effectively 1D?
These questions don't seem to have "obvious" answers to me, and they're only addressed as "assumptions" on this page.
As we rotate the table we know that in some locations a particular leg will be too short (wobble when we press in that location) or become too tall (source of wobble for another leg).
As long as the ground is continuous and does not have any cracks to introduce discontinuity, we know that there must be a location that the leg is the exact length. By the intermediate value theorem the length cannot go from too short to too long without a solution in between.
I first ran across the Wobbly Table when my wife was studying the Borsuk-Ulam theorem. It is fun since you can effectively solve wobbly tables on many patios.
I use it all the time to secure stepladders into stability before climbing them up.
But the table surface might not be level, if my thinking is correct.
As for @Cerium's real-life usage, you have possibility of uneven legs and uneven floor (and discontinuities, like a raised floorboard) so it's obviously not guaranteed, but if the floor is warped and smooth enough, you can try.
[EDIT]: Changed wording
https://people.math.harvard.edu/~knill/teaching/math1a_2011/...
I wrote about it here: https://haggainuchi.com/wobblytable.html
My preferred proof goes through a theorem of Dyson that any continuous real valued function on the sphere attains equal values on some square on a great circle.
IMO this proof is a good example of science taking itself much too important.
Visualizes a proof and talks about special cases and a little of the history.