In my personal experience I did 40 pushups every other day for months and my fitness level and happiness did not noticeably increase. Also did curls, tricep moves with 25 lb dumbbell. And crunches.
Then I got a weight set where I could progressively add weight every week. And ate more and had appetite. That finally made a difference in fitness level and mood.
Bodyweight exercises can be made progressively more difficult, too; it's just that it's not quite as obvious how to do it. But there's definitely a mindset that a pushup is a pushup is a pushup, which is sort of limiting. There are a number of programs out there that offer stepped progressions for bodyweight exercises that provide a similar ladder to freeweight work (working, for example, from knee pushups through, eventually, to one-armed pushups). Not saying that's the right way to go for everyone, but the barrier to entry for bodyweight exercises is satisfyingly low, and for some people, not having the excuse of "I can't exercise because I can't go to the gym" and/or "I can't exercise because I don't have X equipment" is a net win.
Yes you can do a bare minimum but without progress you may lose motivation. After a while you get used to the slightly better feeling and slightly better sleep. May get bored with it only to go back to the occasional workout. That was me. I needed a more noticeable result.
Progressing to complex bodyweight exercises is trickier. Lifting weights properly is technical enough, planche, even headstand pushups weren't going to happen for me. I made every form mistake you can make with weights. Mistakes with complex bodyweight moves halt progress.
Then I got a weight set where I could progressively add weight every week. And ate more and had appetite. That finally made a difference in fitness level and mood.