No, you shouldn't have to cut anything. You should be able to get at it from the inside. My '72 doesn't have a hull liner, so I wouldn't think that your '71 would have one in the way, either. The plywood doesn't have to be the full height of the transom, just a little larger than the base of the motor mount.
Although, taking another look at your pictures, you have a pretty big motor mount compared to others out there. Looks a lot more substantial than what I have on my Mac 25. It seems to take up most of the transom already, in the picture. On the water, you'd probably be fine leaving it as-is. It's just trailering with the motor installed that could conceivably be a problem. But even then, you might be okay with such a large footprint at the mount's base.
Although, taking another look at your pictures, you have a pretty big motor mount compared to others out there. Looks a lot more substantial than what I have on my Mac 25. It seems to take up most of the transom already, in the picture. On the water, you'd probably be fine leaving it as-is. It's just trailering with the motor installed that could conceivably be a problem. But even then, you might be okay with such a large footprint at the mount's base.