If you are training for strength you need your bench. BW pushups offers a resistance of perhaps 40 kgs depending on bodyweight of course. If you can bench > 60 kgs that's not strength training. You can add resistance - but there are limits to how much you can add (20 kgs perhaps) before the exercise becomes ridicoulous. 5s are essential to basic strength training - and you benefit from doing 1RM 3 RM and 5RMs from time to time too. This simply isn't practical do achieve with pushups.
I really don't think this is true.
Many of the old time strong men, and bodybuilders, got brutally strong using the overhead press. The obsession with bench is a fairly recent development.
Yes, these guys probably could have gotten even stronger with the bench in their routines. But do many sports people truly NEED any more upper body strength than say, a 1.5-2xbw press would offer them? Many studies have shown that after a certain point the benefits from strength training for athletes have diminishing results. And considering that if you practice any kind of sport you have limited time to dedicate to strength training, it becomes even less and less of a priority. And especially considering the time taken to increase your lifts by small amounts as you get into this "advanced" level of strength training.
And if you are worried about more chest specific strength, rather than just general upper body strength, what about the other all-but-forgotten chest exercise, the dip? This is also a closed-chain exercise which could be considered to be much superior to the bench press. Also many of the weight-loading issues that you may experience with push-ups are much more easily dealt with.