Richie simply wasn't respected or liked by any of the Capos or Soldiers in the family. Only idiots seemed to show allegiance to Richie, (Jackie Jr., Dino, Sean Gismonte, and Matthew).
He was a loose cannon and making Beansie (a well-respected and well-liked store front) into a paraplegic didn't really help matters either.
Precisely. Junior was a crafty old bastard, he knew better than to back somebody like Richie that nobody respected. Imagine what would have happened if he did back Richie:
Richie whacks Tony, and then what? Was he planning on magically stepping in to become Street Boss? Paulie and Silvio are still around (and Pussy too but he's a rat), so Richie is highly likely to get whacked himself in short order or maybe even get caught for the murder if Pussy let the FBI know to look into it (not that I think he would, since he'd likely want to whack Richie, but still it'd be an option). It doesn't take a genius to figure out that Richie wasn't smart enough to plan this on his own so had to have somebody backing him, and the obvious person would be Junior. So Junior has just started a civil war with what little he has (if he even has anything besides Bobby) versus Paulie/Silvio and the remaining Soprano crew. Barese is a pussy so would likely back the Soprano crew too. So Junior has nothing and is likely whacked and the empire crumbles, either being reformed under Silvio or Paulie or getting swept up into New York (and, with Tony dead, it's debatable if Pussy would have been found out as a rat). Honestly, interesting things could have come from such a scenario, but it wouldn't be the Sopranos that we all know and love

So yeah... there's no way he could have backed Richie. It would have been disastrous for everybody. Richie might have been a stone cold gangster but demanded respect for things from 15 years ago and just stepped on everybody's toes while tossing around his brother's name as justification, ironically in a similar way to his nephew Jackie Jr. (must run in the family or something).