You say it has this play when engaged, but you really don't know that. I'm not being smart. The engagement is through the PTO clutch/brake and cannot be checked "engaged" because that would mean the the PTO is turning. In other words, the PTO is not truly 'engaged' unless the engine is running to generate hydraulic pressure for the clutch. When the rear PTO is engaged, it's a hard engagement through a drivetrain. Are you sure you are seeing a full 3/4 turn rotation? The rear PTO turns at 540 rpm and the mid PTO turns at 2000 because they have different drive gears, but they both engage off of the same PTO clutch. Because of the gear ratio, any slop in the mid PTO will probably be only 1/4 that amount on the rear PTO shaft.
Do you have noise when the Mid PTO is powering something? Are you aware that both mid and rear PTO can be engaged at the same time? You just have to select both levers before moving the modulation lever forward to engage the PTO clutch. 3/4 turn sounds like a huge amount to me. I don't have a mid-pto on my tractor, so I can't give great feedback. Surely somebody else can test theirs and tell you what they find.