This description isn't anywhere good enough to be sure what's going on but the lift Rega uses is a pretty standard part. Look close there should be a couple very small hex screws used for adjustment. One holds the arm lift at the angle you set. Another one holds the whole lift mechanism in place. If this one is loose the whole thing could wobble around as you described. This one is also used to adjust the height of the lift.
With a little work you can adjust both of these to where it will not be possible for the stylus to contact the platter even when the lift is down. Setting it up this way also gives a lot of drop giving you more time to get to your seat while the arm is going down.
With a little work you can adjust both of these to where it will not be possible for the stylus to contact the platter even when the lift is down. Setting it up this way also gives a lot of drop giving you more time to get to your seat while the arm is going down.