duke7595, I had a 1975 JD 1120 for many years. Your 3rd picture of the clutch pedle and the PTO lever is identical to what I had, as well as the two gear shifts - gears 1-4 on the right, HI, LO, reverse and park on the left. Depressing the clutch part way( going to say half way, could be more) allows the operator to change the transmission gears. Depressing the clutch fully to the foot deck allows engaging or disengaging the PTO. You should feel the different stages engaging as you depress the clutch. As others have indicated there is a clutch problem. Maybe your mechanic attempted to install a "will fit" single stage clutch assembly instead of original 2 stage Deere parts. My neighbour had a 1020 a few years older and his operated the same way.
I went back to your original post. Someone here has mentioned that the 1020 was available with a single stage clutch or no PTO - news to me for sure. I always thought that Deere was one of the leaders in promoting of "live PTO" or the 2 stage clutch. Anyway, suggesting that you have to pre-engage the PTO then start the tractor is just plain wrong. You have a clutch problem that the mechanic has failed to fix for you. Find a mechanic that has worked on a green tractor maybe once or twice.