After that, Knuckle Duster's words were misinterpreted, and he decided to take out any hero who was only in it for themselves, in an attempt to get people to understand the problem. His ideology? Correct. His methods? Not so.
His ideology is not correct at all, he's clearly deranged. People wanting reward for their work is never treated as a bad thing in universe. Only Stain acts like that.
He wants people to work themselves to the bone like All Might and desire no reward. He doesn't care how unhealthy that life-style is, look at All Might. He had nothing, his only friend was a policeman he met after defeating All For One.
Imagine if Horikoshi kept him as a spy for AFO, All Might would have no friends.
He didn't gain connections until he met Izuku and became a teacher. Which has nothing to do with Stain's opinion or belief on what makes someone a hero.
Tensei's only motivation was to help people, because he thought that was the coolest. His public motto is to rescue people as quickly as possible and to have solid teamwork. Yet despite that Stain saw him as a fake hero and crippled him for life, despite all of the good he's done.
Tensei was a true hero, he was one of the most kindest men in the series. Stain doesn't care to actually check if these people are "true heroes", he just takes anyone not named All Might down. Only letting Izuku and Shoto go because they showed promise the second they appeared in front of him.
Basically, if you don't instantly impress Stain you're a weak fake hero that needs to be culled. Tensei's first words probably weren't All Might level so he got cut down.
Note: He let Tomura go as well. Tomura was still growing and his convictions weren't fully realized, Stain wanted to see what he'd grow into. Though he admits he may have to kill him in the future, he still just casually let a villain go because they both agreed on destroying the present society.
The series never showed us money grubbing Heroes being a problem. No, I don't count the mass retiring of Heroes from the All-Out War aftermath as this. These people were not trained or prepared to deal with the entire country shutting down and having all of the civilians hate them for not stopping it.