Here's the Rangers Schedule in January that you posted.
The bolded teams are currently playoff teams. I've indicated the games we won with a green 'Win' beside it. Three of our wins came against playoff teams. Three. Of those playoff teams that we beat, only two of them are in the top-4 of their conference. So unless you're counting on playing weaker teams all the way to the cup, I'm not sure how any of this really fits into us being legitimate contenders for a cup.
People see a string of wins and they think suddenly this team can make a run at it. In reality, we're beating mediocre teams during a very friendly travel schedule. There's a very distinct difference between "competing" and being legitimate challengers for a cup. Whenever that is brought up, sure enough people come forward with the "What about the Kings or Flyers!?" Basing your hope for success around a statistical anomaly is something coaches champion to their players when they're the underdogs. It's not something you build a team building strategy around.
I don't want the team to lose. Quite the opposite in fact. However, I don't want the team to make important team building decisions around a recent run of success against lesser opponents. Sure, we could have a great March, but if we're not beating the teams that we're going to have to beat in the playoffs, then the month-to-month record doesn't really matter.