Have you checked the alignment yet? Your initial problem description says it pulls one direction, the same direction, only during braking. Your video shows the steering moving around during driving of all kinds. If that's your video, your initial post did not properly describe the issue.
Alignment issues combined with wide (especially wider than original spec) front tires can cause this. It's sometimes called tramlining. If there's excessive toe out the car will want to follow every rut in the road as well as road crown in general (e.g. "pulling to the right" most of the times you use the brakes). Particularly if it's an M3, the previous owner might have done this intentionally as 'toe out' increases steering response. Are the roads were you live rutted by heat or heavy traffic?
Part of an alignment by a good shop is a pre-alignment check. They will pry on every bushing and ball joint and make sure that it is at least in good enough condition to take an alignment, and it's an extra set of eyes on the car for no extra charge.