Note that gyroscopic precession is a reaction to a torque, not a lean angle. If a bike is leaned over and in a coordinated turn, there is zero torque on the front wheel, and there is no gyroscopic precession reaction.
In the video one of the reasons a bike self-steers in the direction of lean is having the front tire contact patch behind the point where the extended steering axis intercepts the ground / pavement. The distance from the extended axis intercept point back to the contact patch is known as "trail". The video also mentions having weight in front of the steering axis also causes the front tire to steer in the direction of lean. Note that this self-correcting reaction to lean angle steers inwards a bit more than what is needed for a coordinated turn, and tends to return the bike to a vertical orientation.
As speed increases, the gyroscopic reaction increasingly dampens (opposes) the self correcting steering geometry and on motorcycles at high speeds (100+ mph == 160+ kph), the gyroscopic reactions dominate, and the tendency transitions into one where the current lean angle is held as opposed to returning to a vertical orientation. If the bike had knife edge tires, at these high speeds, the bike would be falling inwards at an extremely slow rate (called "capsize" mode), but with real tires, the usual tendency is hold a lean angle.