I assume you mean sideplay, cuz
at the top they all have vertical play, that's normal. Excessive sideplay is not.
The factory spec is .000 to .003. I have seen more than .003 total radial play and the cars run just fine.The factory service limit is .006
Obviously excessive sideplay changes the dwell which changes the timing .
But the thing is, that top part does not spin on the shaft, but only rotates back and forth about 20 to 25 degrees, from idle to all-in which can be as high as 4000rpm.
But when you switch to electronic ignition, the radial play becomes somewhat inconsequential. The dwell is now set by the ECU, and the reluctor gap can vary widely and not disrupt your ignition, all it does is signal the ECU when to fire.
Wore out after a day?
The installer is supposed to lube the pointcam with the special grease made for it, and he is supposed to clean the contacts with contact cleaner as well as whatever tool he uses to set the gap or dwell, and to oil the top of the shaft under the rotor, which will tend to center the cam and decrease the sideplay some. There is a felt wick in there that will hold quite a bit of oil.
And the car owner is expected to maintain this system. Probably twice a year, depending on the accumulated mileage.