In his encounter with Crablante before he started training, Saitama is hit by him and thrown away like 30 meters or more, and then hit again 2 more times with those gigantic claws Crablante has. If Saitama was a regular human, how could he support this beating and still stand like it was nothing and fight back?
Just saying, the hundred push-ups stuff is what Saitama thinks is the reason for his incredible power. And remember Saitama isn't exactly the smartest person to say the least.
I heard somewhere that the reason Saitama is so strong is due to the fact that he broke his "limiter". Now there isn't much information on this but I imagine that it works like this: every person has a limiter based on who they are. This limiter determines how strong they can get mentally and physically. So in Saitama's case he not only trained really hard but also gave everything and when I say everything I mean EVERYTHING.
Before he became super strong he had no job, no family, no friends, he basically had nothing to live for. As Crablante said "I see that you have lifeless eyes." "From one pair of lifeless eyes to another I will let you live." (Something along those lines, but you get the point.) Back to the limiter. What I am trying to say is that Saitama gave it his all, his desire so strong that he broke the barrier that stops everyone from becoming too strong.
This is my main theory on why Saitama is so strong. Now there's this other theory about him eating all his enemies to become stronger and stuff, but that's a whole other story. (I mean no offence to anyone as this is only opinion on this topic. The people who make One Punch Man probably have this completely different explanation on why Saitama is so powerful.)
Before training, at the time of fighting with that Crablante, Saitama told that he wants to become a hero who can kill any monster with one punch.
I think there is someone/something which grants power to humans or awakens their power according to wish and righteousness. After fighting with Crablante, we can see Saitama inherited/awakened his power which is shown with his eyes being orange color. After that, he trained a lot and became a super powerful one.
Yeah, of course, he had to have had some capabilities to be able to attempt that regimen for over 3 years, but saying that he “only defeated a tiger-level” before he started training is belittling the hell out of what happened, no way he’s a high C/Low B-class hero.
To beat a tiger-level disaster (in this case Crablante), you have to either have 5 B-class heroes or 1 A-class. So, he would have been in a higher class since then.
If Saitama was able to even attempt to do his physical training regimen of 100 pushups/100 situps and 10km run, he had to be physically capable to a certain extent to even attempt something as vigorous. So even at that time, he might've been as strong enough to be a high-ranked C-class hero or low-ranked B-class hero perhaps. Also, Crablante is only a Tiger threat-level monster.