Because when Junior first took over the family Tony met with the other Capos and he said Junior had New York behind him.

I guess that's true...
Carmine and him had nice talk at that sitdown about ginny sack joke when they were using that special phone sil gave them.
Junior was smart (maybe not as a boss material, but more of a consigliere) and old school mobster so i'm pretty sure they backed him...

He had the Mangano and Tarese families behind him, while also having a good relationship with Carmine, so three out of five families pretty much makes New York behind junior.

Mangano used to be the name of the Gambino family. That should be obvious enough. My takeaway is that Uncle Joon had the mustached petes behind him, for whatever that was worth by 1999, while Tony had the young turks behind him. I'd include Carmine Senior with the latter considering his dismissal of John Sack's complaints about Ralphie insulting his wife.

Johnny Sack said that the Lupertazis and Sopranos go way back. Also, John didn't know about Tony secretly running things until after the captains already agreed to it, so the Lupertazzis were supporting Junior when Tony said that.

Lupertazzis ultimately support money, and Tony earns more than Junior. Probably tenfold. This is what Larry Barese tells Tony when they're having dinner and discussing where New York stands on Jersey leadership Carmine nearly killed John Sack, because John wanted him to kill Ralphie over the insults toward his wife, which would have cost Carmine a ton of money. I think the writers on this show did a good job of showing the conflict between money and tradition, and how in modern times, money usually wins out.

Junior definitely had NY behind him at one point. I'll put it this way: Tony did NOT want to have to go to war with his uncle. I would tend to believe Tony when he said Junior had NY behind him. Junior goes back in the Family to the 1950s, if not even the 1940s. That's almost 40-50 years prior to 1998/1999, plenty of time to make lots of heavy contacts and connections. Carmine was Boss of the Lupertazzi from at least the 70s or 80s, when Tony was not really a major player. Junior would've taken a larger hand in things as Johnny Boy was dying, too.

Those stats Junior could claim would only bury him deeper.

The more that could be taken away from the weak old regime, makes it a more easy move for the new young regime to claim.
Tony took the backdoor, smooth and most lucrative route to the Top. Which actually was the best move since the timeline Is not exact.

Jackie Aprile was boss when the story begins. Before him, it was Ercole Dimeo. If Carmine was doing business with New Jersey for a long time, then DiMeo and Jackie Aprile had to have been his primary liaisons. I can't think of any scene throughout the show's run where anyone in New York speaks highly of Junior's effectiveness. New York despised some of Tony's narcissist ways, but they knew that across the river, he was the only game in town.