As stated Monahan gets away with a lot more on a rebuilding team then he would with a contender (Jets or otherwise), playing offense with little expectations on playing good D obviously would help anyone's offensive numbers (playoffs is even far tougher then reg season). Being played on a rebuilding teams #1PP isn't something he would get on any contender team likely so again hurts his output offensively (at the very least his PP1 time would be going way down). Counter to that will be he's playing with better line mates, likely is true if he plays a top 6 role but not necessarily true if he's playing a contenders 3rd line.
Monahan is a lesser option/cheaper option compared to some better options being discussed so I'll give you that, but if a team truly wants to contend and put their best foot forward Monahan isn't the guy you go after, he's the guy you settle on when everyone else is gone or unavailable. Can't imagine any Habs fans would even disagree on that point or at least would acknowledge that's likely a fair point.
From a Jets POV, my hope would be as per the rumors is the Jets are aiming higher then Monahan and I hope they get their guy. Plus maybe add a top 4RHD