Is the scooter's right handlebar (accelerator) loose?

That's not sloppy, this phenomenon is called free travel.

This is a design requirement to prevent the throttle from closing improperly.

Isn't that how cars are designed? Although the control methods are different.

The throttle control cable of the scooter controls the carburetor throttle, and the throttle is equipped with a return spring to keep the throttle closed all the time. If the throttle knob has no free clearance, how to keep the throttle in the closed position? If this gap cannot be guaranteed, there will be faults such as excessive throttle opening and high idle speed.

Because the throttle handle of motorcycle works in rotation, its radial clearance is to make it rotate more flexibly, and the axial movement is the throttle free clearance I mentioned above.

This situation is normal. If the radial amount is too large, you can consider replacing it, but under normal circumstances, there will be no problem of falling off during driving. If the free clearance is too large, the clearance can be changed by adjusting the throttle cable.