This is not a new argument against SR. It has made the rounds on a number of crank websites dedicated to overthrowing SR and it probably has its origins in some textbook somewhere. It is likely that someone who was not able to work their way through the problem decided to try to use it as an argument against relativity instead.
The fundamental unit in SR is the spacetime distance. This distance is measured between two events and it is always the same number in every frame of reference that uses the same units ( centimetre, kilometre, mile, etc.). The spatial length and the temporal length between two events will change depending on frame of reference used, but the spacetime distance will remain the same.
SR is also event-preserving. That is, if a certain number of specific events happen in a scenario in one reference frame, then that same number of specific events will occur regardless of the reference frame we use to describe that scenario. In the use of SR, one often refers to any spacetime point as an event, so I use the terminology “specific event” to refer to the result of some physical interaction, such as the collision of two objects or a timer going off.
So if we are provided with a distance between two events and we are told that a certain number of events happen between these events, we can be sure in SR that any reference frame that we use has the same number of events. Additionally, if we are given exact details for the location of these events in one reference frame, then we can find the location of these events in any reference frame.
What we can’t do is claim that the spatial distance that these events span is the same in one reference frame as it is in another. We also cannot claim that the temporal distance that these events span is the same as one reference frame as it is in another.
In the scenario at hand, we have a wheel that spins in contact with the ground. One point on the wheel performs a specific event 1000 times. That is, one specific point on the wheel hits the ground 1000 times. These events will happen in every reference frame. In one sense, they will happen at the same location in every reference frame. However, the coordinate assigned to each location depends on the frame used.
What the scenario above really shows is that relativity demands that we cannot expect things like regular counters to behave the same in every reference frame. In some they will count faster than they do in others. This is hardly surprising, since these regular counters are effectively clocks and SR tells us that clocks behave differently when they are in relative motion.
You can work this out for yourself or read up on relativity theory in a good book.