The car in the video is not likely FWD as I saw it. But, if it is FWD, "Torque Induced Steer" is a common problem cause by a positive NET scrub radius. Sure, the geometry indicates a negative scrub radius, but the static + dynamic pneumatic scrub added by some tire constructions (Tire Mx / Fz ) will cause the "twitch" as you describe. Basically the P-Scrub is quite a bit larger than the geometric value and is the opposite sign. A -15 mm scrub radius plus a +50 mm P-Scrub is a positive number. Turn the tires around on the rims for a Big Science undertaking. It is common for tires of the same size, rim, pressure, but different brands to have very different P-Scrub values. This is a real problem on high hp FWD and 4WD vehicles.
As for caster induced understeer, caster increases net tierod load. Tierod loads add compliance from steering gear mounts, rack/pinion separation forces, hydraulic valve reaction rotation, steering column isolator twist and steering intermediate shaft U-joint displacements.
In just about all cases, peak tire Mz occurs before peak Fy. Mz is a contributor to steering effort, and a component of "road Feel" is its derivative (gradient). Most drivers will operate up to the point where this gradient goes to zero, even though more Fy is available. BTW, the Mz induced compliance component to vehicle understeer is also going away, so the car's 'balance' will change.
So, adding caster moves the apparent peak steering moment gradient upward on the slip angle function so it coincides with peak Fy (if done properly). But, this adds understeer which lowers the vehicle's max Ay capability.