I hate to break the bad news to you, but if you suddenly increase your pushups all at once.. like doubling them, then you might start to get really sore, so you may or may not be able to continue to increase, or if you do too many sets in a row, you might get more sore from that, so even though you might improve your pushups, you also might have some days that you are even more sore than you had been on previous days.
Maybe this will all work out for you, but sometimes there can be advantages in spreading out your pushups throughout the whole day, especially if you are going to continue to want to do several sets in each day and try to continue to do them every single day (even if some days might be higher numbers of pushups and other days might be lower number of pushups, but as you build, you still should be able to continue to increase your overall quantity of pushups, as long as you do not end up injuring yourself or otherwise overdoing it). It is like investing into bitcoin. You want to be aggressive, but not to overdo it otherwise you might end up injuring yourself..
Yeah, I completely agree with you and must have said that out of experience, increasing your pushups all at once doesn't only make you sore but it also make you to have the feeling of dout whether or not you might get to your previous high which might apparently weigh you down and you might end up doing below expectations while focussing on how to get to your previous high number and this can actually lead to being overly aggressive and you might also end up being injured on trying to do that.
Spreading out your pushups throughout the whole day are too advantageous as it gives you a good sum total which off course has impact to your body after having some rest on different intervals making you body to get use to certain range of numbers on different days given you the opportunity of gaining more stamina that will enable you to gradually increase your overall number in different days.