No one said he couldn't be a villain

See reality for what it is

"You need to see reality for what it is, kid." All might said and he left the roof.

Then that was it, despite being told ever since Izuku was discovered to be quirkless that his dream of being a hero was hopeless, being told by his idol, the number one hero all might, a person that made the impossible possible and had unknowingly motivated him this far...

Just being told he should give up...

Give up on everything he wanted up until this point, just because it was unrealistic?

No, it wasn't that it was because he's weak because he doesn't have a quirk!

Katsuki had been right, Izuku was a weakling, a pebble on everyone's path, he wasn't capable of anything like everybody else, he didn't have a quirk like everybody else, no one believed that he could become a hero like everybody else, not all might, not even his own mother ever held any faith in his dream, after they knew what he was!

Quirkless, unlike everybody else...

It seems as everything Izuku wasn't was because of that one simple fact. He was born quirkless and so wasn't able to be a hero everyone had told him countless times.

"Haha everybody was right I'm just useless. Not even the number one pro hero has any faith in me..." Izuku said to him self. Tears beginning to form in his eyes as he screwed his fists determined not to let them spill.

"I can't be a hero it's just impossible you've got to be realistic Izuku you can't be a hero without a quirk, you can't save anyone without a quirk, you can't even protect your self from Kacchan without a quirk!" He sobbed wiping the now falling tears as he fell to the ground.