Even with axial play it should be possible to make a straight cut. This is a rubbing tool and you still don’t put on coolant. Aluminium you either hog very aggressively roughing, then no coolant’s needed, or skive it slowly with a sharp tool under coolant.
I am a professional turner and I’m telling you not to play around. Take it seriously and adopt the geometry engineers have contemplated. None of the angles of the tool in the video is correct, top rake, side rake, lead (or approach). Get away from carbide when you can’t have the necessary speeds.
To set the tool post simply loosen it, push it up to the chuck, claws removed, tighten.