Reason - 9" grinders operate at a lower speed, and the disks are intended for use at that speed (typically around 7000RPM). A 4 1/2" grinder operates at around 11,000RPM so using a worn down 9" disk is obviously expecting the disk to operate faster than it's design speed which could cause it to break up.
Your 'reason' is not correct. Please look at the markings on the disk and think through what they are actually saying.
https://www.kissafety.co.uk/abrasive-wheels-understanding-the-marking-system/
Any maximum rpm given on an abrasive disk relates to its original (unworn) diameter. In the context of the present discussion, a more useful (more critical) figure that is also given is the maximum surface (peripheral) speed. For a bonded wheel (angle grinder disk) it is usually 80m/s.
Using your own numbers, a 9" disk spinning at 7000rpm actually has a higher peripheral speed (82m/s) than a 4 1/2" disk spinning at 11 000rpm (65m/s).
Putting a 9" disk that has worn down to 4 1/2" diameter on a 4 1/2" machine is no different to putting an unworn 4 1/2" disk on the same machine.
In a way, your logic is upside down: to work efficiently, the disks need a cutting speed near 80 m/s. That is why your tiny Dremel wheel goes at >20k rpm, so its perimeter is still moving at near to 80m/s.
Huge wind turbines spin very slowly but the toy windmill you buy at the beach spins very fast. The peripheral speed of the blade tip on each is comparable.
The rpm of the front wheel of your Raleigh Chopper has to be considerably higher than the back wheel in order to maintain the same surface speed along the road.