Here is a strategy to work towards your first strict push-up.
It sounds counter-intuitive, but weighted push-ups can help you get that first strict push-up. The body can handle much more load during the eccentric, or lowering portion of movement. See pictures below.
The Eccentric Phase:
This is beneficial because the majority of muscle breakdown, which causes growth happens during the eccentric phase. Basically more stress on the muscle = more muscle (as long as you recover effectively). Adding load is easy, just place something with weight on your back. Obviously a barbell plate is optimal because it is easy to balance and you can easily track the weight used. If you are doing this at home, get creative, you can use a phone or text book, a bag of potatoes, or have someone push down while you resist.
To come up, simply reduce the load before pushing up by removing weight off of your back or coming up from the knees. By doing weighted push-ups, you also get the added benefit of core strengthening since you will be holding yourself in a plank position throughout the movement. Just make sure to keep the core tight!
Now drop down and give me 20!