There are a fair number of exercise programs that rely on one movement. Just one exercise - although they occasionally sneak in variations.
Why do these in the first place, and why do these work?
Freedom from Choice
If your workout has you doing 100 pushups a day, or kettlebell swings three days a week, or 500 bodyweight squats every other day, you know what your workout is.
You don't have any guesswork. You don't have any planning to do. Nothing to wonder about or second-guess yourself over. You have the one movement to do. You'll have exactly zero stress over the workout planning as long as you let go and trust that the exercise you chose is the right one. All you need to do is execute. This eliminates the "analysis paralysis" problem where you just don't act due to too many choices in front of you.
If you workout is just one movement, you can't help but work it consistently. You will improve if you work at that movement with any significant level of effort. A one-exercise approach means you will keep hitting that exercise over and over, and thus bring some consistency to your workout.
You can't fake progress
Either you get more sets, more reps, or more weight - or you don't. You can't switch exercises and tell yourself you're inducing "muscle confusion" or "shocking" your system into growth or whatever. Either you get more pushups, or more swings, or squat more weight for 20 reps, or you don't. There are no two ways about it.
That's basically why these programs work. You're free from choice, and have only a simple road ahead. It's a good way to progress if you can find a solid full-body movement, and just stick with it.