I think the descrepancy Holmes sees – & incriminating the gardener – is not whether the bike and rider fell to the right or straight ahead. It’s simply, that the rider wouldn’t have fallen off the bike to the left. As mentioned above, when bike handlebars are quickly jerked (by hitting a rock or on purpose) without the rider preparing for the turn by leaning, the bike will initiate the turn, but the rider’s momentum will usually cause him to go over the handlebars and land on his noggin, more or less straight ahead.
I concur that the gyro effect of the front wheel is probably not an important effect. The reasoning for it tho would be based on that experiment all first year science students do in Physics 101. This is how I recall it works. You stand on a big lazy-susan, and someone directly in front of you hands you a spinning bike wheel, which you hold by both handles, each attached to alternate sides of the wheel axle, so that the wheel is spinning vertically in front of you. Say the direction you are facing is North. Now you are asked to turn the handles so the wheel is pointing West. You can turn the axle counterclockwise ok to try to point the wheel to the West, but the lazy-susan you are standing on will turn in the opposite direction, clockwise. You’ll end up holding the spinning wheel now pointing to your left, but since you turned too, the wheel will still be pointing North. And you’ll be facing East.