Obviously, I'm not Cathe, but I think 8 push-ups on your toes is a great start!
I know what you mean about the P-Us at the beginning of C/T. I usually put more recovery time between the sets, because I want to work on doing them all on my toes (and I can in other workouts).
I think you have to decide what you want to work on most: muscle endurance and recovery? Then do them as they are, as many as you can on your toes, then go to ONE knee (maybe even on a balance disc) before going to two knees). And work on doing more on your toes as you progress.
Or would you rather work on muscular strength (as I do)? Then take longer breaks between the sets to allow for more recovery, and work on shortening these breaks as you go along.
Push-ups really call on a lot of muscles, and the coordination of several muscle groups working together. I find that they really require scapular (shoulder) stability and core strength. Other chest exercises usually don't.
You can work on both of these by doing planks in the 'top-of-a-push-up' position, and gradually increasing intensity (by staying up longer, by lifting one leg, by lifting the leg then bringing it in and out to create a stability challenge.