While it doesn't look like it is going to happen, here is the rationale:
Billups is in the last year of his deal, and he's lost his quickness and athleticism. He can still shoot, but he's a defensive liability and can no longer penetrate.
Paul is the best pure PG in the NBA, and is a huge upgrade defensively. His skill at getting the ball in the scorer's hands would be huge for the Knicks.
A'mare is a much better offensive player than Chandler, without question. But what the Knicks have been lacking is a defensive force in the middle. Chandler has the combination of rebounding/defense that could really help, and is a big upgrade over A'mare in those facets of the game.
I think it would be largely beneficial to make those moves. The best way to build a team in the current NBA IMO is with a great distributor(Paul), wing scorer(Melo) and a dominant defensive presence inside(Chandler).
It's part of the reason the Heat had such problems last year, even with having arguably the 2 best players in the NBA. Wade and LeBron's skills are largely redundant. They both need to have the ball in their hands. They got by on pure talent, but a better constructed team will beat them.