It's called frequency method and it fucking works but it shouldn't be max pushups. It should be sub failure sets, they should be very very easy sets, about 50% of your max number.
And yes you should be trying to balance the pushing and the pulling. I would suggest inverted bodyweight rows instead of chins though to balance horizontal pushs and pulls.
If you cant do frequent rows, do ladders in the morning and at night. A ladder is where you do 1 rep, rest the amount of time it would take a partner to do a rep, then do 2 reps, 3.....until you can get the number of reps for a given set. Instead of going back down. You start back at 1 again. Allows you to add volume easily and because your hardest set is followed by your easiest, it allows in set recovery to a point.
Do a bunch of band dislocated and if you have a suspension trainer or rings do rear shoulder pulls(y deltoid exercise) or face pulls.