Right, but is it better at buidling strength than a 5 set program with rest days? Is it less likely to get me hurt?
Let me give a fuller answer to this. I am somewhat confident that, for a given volume of work, a beginner can build strength faster without rest days. I am extremely confident that a person is less likely to get hurt without rest days.
You can build strength in two basic ways: (1) your brain can learn how to recruit a greater number of muscle fibers more efficiently, and (2) your individual muscle fibers can physically adapt to improve their contractile strength.
With a beginner, early gains are mostly in the first category. In this sense, strength is a skill, similar to playing piano, improving your jump shot, or learning to juggle. Skills are generally acquired more efficiently with lots of short practice periods rather than a few long periods. If you want to spend six hours a week learning to play piano, for example, you're much better off practicing for an hour a day six times than you are practicing two hours a day three times. Better still if you practice three times a day for 20 minutes each.
You get diminishing marginal returns within each practice session as you start to get fatigued and are no longer fresh.
The same goes for practicing your pushups. When it comes to neurological (as opposed to muscular) adaptations, if you're going to do 300 pushups a week, you're better off doing 50 a day six times rather than 100 a day three times. Better still if you spread those 50 out over multiple workouts a day day instead of doing them all in the same workout. You'll be fresher for each rep that way.
The benefit of spreading things out is more obvious when it comes to avoiding injury. You're most likely to get injured when you're fatigued. You can minimize fatigue by spreading things out as much as possible. I'm not as sore as I expected to be after walking 50 miles, but I'm still walking with a bit of a limp because the area behind my left knee is pretty tender. I can say with close to 100% confidence that injury is much less likely walking 10 miles a day for five days than it is walking 50 miles in a day and then resting for four. Same principle with pushups.
(Once your beginner gains are used up and you're mostly improving strength through muscular adaptation rather than neurological adaptation, there's a good case to be made for intentionally fatiguing your muscles a lot more instead of spreading everything out as much as possible -- which means concentrating your volume into fewer, more intense workouts. But even extremely advanced strength athletes can do very well on a protocol with no rest days. The national teams in the Iron Curtain countries famously did quite well in Olympic lifts and power lifts while training every lift every day, no rest days. It's possible that they also benefited from taking extra vitamins.)
There's also another factor, which likely varies among individuals:
If not including rest days isn't the biggest mistake people make with exercising it is right there.
I think the biggest mistake people make with exercising is not doing it enough.
For me, personally, it's
extremely helpful for me, if I'm going to stick with something, to make it a daily habit. If I take days off by design, I'm much more likely to take more and more days off and eventually drop it. This is my own issue that won't apply to everyone, but I don't think I'm entirely unique in this respect.