I see the direction you're heading. But I have a few concerns.
First, the steering mount travels in an arc, not a straight line. The mount is about 10 inches from the pivot point.
Second, it seems like it would be hard to have a manual override with that design.
Third, I'm not sure how it would work with tipping the motor up while the boat is being towed.
I took a further look at what was left of the old steering system. Attached to the steering wheel was a small gear that turned the big gear that pulled on a cable. The big gear probably makes only a half a turn. It has 2 placed where the cable comes out. If I had the cable & other parts, I would connect a geared motor to the steering wheel. Alas, that is not the case. But perhaps I can use the same concept.
-Kevin