Why did Hillel not turn a convert away three times, not even once?
And why did he convert them straight away, when if seen by todays standards, they're the most terrible candidates.
They were either not serious or were heretics!
(the guy on one foot was not really serious, the guy that wanted to be kohen gadol wasn't serious, they'd be laughed out of any beis din or even any pub as even a frivolous person would see they are frivolous. And the guy that denied the oral torah - which is pretty heretical).
I'm looking at sefaria.org, Talmud tractate Shabbat 31a
The talmud uses the word גייריה which according to the translation seems to mean he converted him.. And it says later that the three (converts) gathered together.
I notice that on Shabbat 31a there are three incidents where Hillel converted a gentile immediately.. and the three gentiles showed afterwards that they were sincere..but not necessarily at the time. One was the famous 1 foot guy. Another was one that immediately before conversion, wanted to be the high priest so he could be dressed like the high priest(so doesn't sound like a serious guy). And another that immediately before conversion, denied the oral torah(really a heretic!).
Also the one foot guy didn't even ask to be converted.
The guy that wanted to be dressed like the kohen gadol wanted to convert, though for the most ridiculous of reasons.
And the guy that guy that didn't believe the oral torah wanted to be converted under a condition that wouldn't really be acceptable to any orthodox authority today - denial of the oral torah!