I am no genius and not even a decent mechanic ... but I have been rising for more than five decades and doing pretty much all ny own bike-building and maintenance except wheel-building for about 35 years .... I don't want to disagree with your friend who says crank wobble is okay .... but NOT ON MY BIKES.

If you have ANY play in the BB you are in trouble. The spindle is harder than the cranks and will stretch the holes. The bearing can fail, and the spindle will eat the outer cups. (been to all those places, done all those things.) The whole unit can wobble and eat the threads in the BB shell, which can be a real bear to repair .... basically you need to buy a rescue BB.

You should be able to move your cranks in one way only---spinning. Any movement in any other direction indicates a big issue which will only become bigger.

One thing about a seized BB---you'd better have a chain-break tool because otherwise you can't roll the bike home. I have actually rigged a sling out of a spare inner tube and a bandana to lift the rear wheel because carrying the bike a few miles home wasn't happening, I have two loaded panniers I would have needed to carry also, and I needed the bike. Five miles with 50 pounds of bike around your neck leaves lasting impressions.