These are all fair points, yet even with a lot of small sets a person may still feel overall worn out and even quite sore the next several days, depending on his own ability to recover including age, physical fitness status, diet, sleep and other exercises that he might be doing (other kinds of stress in his life).
That's very correct.
Dividing one's push up into smaller unit in a day won't stop the burn and soleness in one's body,
When one gets tired.
That is why it is advised to get enough rest when you are tired.
The factor's you mentioned which are age, physical fitness, diet, sleep and other factors are key To determine one's recovery. The level of one's recovery is determined by the person.
So it is advised to give yourself enough time and rest before starting the push up again if you noticed you have not rested enough.
Even though dividing the push up into small units a day won't prevent the burn and soleness it helps prevent total breakdown before meeting your target.
It will be nice not to discuss only the negative impact of push up but to always emphasize on the positive impact of it and the high health benefits.
It is very important to rest and listen to once body when doing push up however we can't deny the satisfaction we feel each time we do what we believe we could not do.