Rear wheel is loose and wobbles at even the tightest setting
It's hard to see from the video exactly what is loose. However, you are presumably tightening the quick release properly so any looseness in the wheel must be from the bearings in the hub.
I see you have a 7 speed 'cassette'. It's quite likely that you actually have a 7 speed freewheel and relatively simple cup and cone style hub bearings. If that is the case you can attempt bearing play adjustment yourself.
In cup and cone hubs the axle is threaded, ball bearings are held between 'cups' in the hub and 'cones' threaded on the axle. Bearing play is adjusted by tightening of loosening the cones.
Here's an RJ the Bike Guy video where he's adjusting bearings after replacing a bent axle. It's a front wheel but it should give you and idea of what is involved. You can search for other videos or guides of course.
You'll need some special thin wrenches to adjust the cones and locknuts.