To be clear: Fan is attached to fan clutch. The fan clutch is attached to fan bracket, with a pulley in-between the two. You can not actually see the fan bracket, in above video. But you can see the pulley belt rides on, where fan clutch attaches to fan bracket.
Fan and the fan clutch itself, may wobble a bit or appear to, until fully warmed up. Watch the video in post #4 very close. Focus on the pulley back of fan/fan clutch, that the belt is riding on. You'll see pulley and belt jump/wobble. That pulley should never wobble at all. It's wobble, is 98% of the time, a failing fan bracket. 2% of the time, loose nuts (4) on studs (4) of fan bracket. Those 4 nuts hold, fan clutch & pulley to fan bracket.
How I check fan bracket, while in vehicle.
1) Visually, as seen in video.
2) Sound. They should run silent. Since we've have belt noise and other bearing noise (idler, tensioner, alternator, vane pump, compressor). I use a mechanic stethoscope, place on fan bracket back of pulley/belt area.
3) Grab pulley and feel for any lateral or radial play, with belt and engine off.
During timing belt service, while fan bracket off the vehicle. I check for sound, play and look for any oil at back seal. Any oil coming from seal, I replace fan bracket. Even if feels and sounds good.
Toyota Aisin fan bracket.
Aisin fan bracket.
I replace, even when oil just beginning to weep from sealed bearing of fan bracket. Dust, sticks to oily surface. Not worth, seeing if can make to next timing belt service. So I replace.