Keeping his real name ~ Spoiler

Why the hell would he start a new life, after faking his death, by keeping his real name???
The movie is good, but that part is only there to give us, tv audience 100 years later, a clever unexpected twist.

Ok they had no wikipedia back then, but even Ketchum, years later with no info on the matter, remembered the name McCarthy from some place...
Billy was a famous wanted man, imagine this guy showing up here, right after his "death", WITH HIS SAME REAL NAME, age and face, and nobody makes the connection???? Come on!

And even if he was that lucky, why ever risk it to begin with?
Pat gives you the chance for a new start. Change your friggin real name to anything else, man!