Sure you can do that, and that's most likely what the last shop you took the bike to did as well.

The crankarm is most likely toast and the wiggle will return. It's highly critical on the very 1st installation that you torque the two pinch-bolts incrementally. And get their final torque evenly balanced as well. As with square-taper cranks, you pretty much have only ONE chance in the very beginning to install the crankarm correctly. Once it loosens and you ride on it for more than a minute, it's buggered permanently.

I've ridden miles one-legged with my left-foot resting on the chainstay to save a loose crankarm.