My old house had a fan that shook because the ceiling box was just nailed to a joist. I grabbed the base plate while it was off and gave it a solid wiggle... whole thing moved. A proper fan-rated box is screwed into blocking wood between the joists. If you can move the mount by hand, that's your problem right there.

Four hundred bucks for a useless jig?

Klein Tools are the standard for a reason. That extra cost up front saves so much time and hassle down the road. You just never have to worry about them failing on you during a job. It's one of those things you only cheap out on once.