Yes it should, the constant dumping of the puck during penalty kills is tedious.
Okay...so people are complaining and saying it would slow down the game too much...so how about THIS?
Okay so they change a couple things around.
1st.) On a PK, the icing line is moved back to the defending team's blue line. So they can ice the puck as long as they get past their blue line first.
2nd.) If a team DOES ice the puck from inside their defensive zone, they are unable to change players on the fly until the offensive team brings the puck back into the neutral zone.
So it will pretty much be the same as it is now except that teams can't change lines if they ice the puck from their defensive zone.
I never understood why a team who takes a penalty was given an advantage in the first place. Hate how teams can just ice the puck with no consequence. You took a penalty, you're short a man and you should have to play like that. I would like to see short-handed teams get called for icing, it'll force them to actually play hockey rather than smack the puck as hard as you can. I'm sure there's arguments for both sides but i would like to see this idea at least given a shot during pre-season or something to see how it actually affects the game.
I feel it's safe to assume that the majority of the posters who agree with the OPs (unsurprisingly) unoriginal premise have never played hockey at a competitive level of the sport, never having felt the cardiovascular burn of a full minute+ PK session when hemmed into your own zone.
Players already expend huge amounts of energy as a defender during a single penalty kill. Being unable to ice the puck and go for a change would translate to a situation where the entire team would be potentially at a disadvantage for the remainder of the game, whether a goal was scored or not on the first powerplay. The initially penalized team will likely commit even more penalties due to tiredness/lack of effort from a single PK than the other non-penalized team.
Also, in that scenario, diving also becomes more likely in an effort to draw penalties, becoming an even bigger issue than it is now, as well as with "make-up calls." Penalties are not handed out entirely fairly - human judgement (and error) is at play via referees. In truth, the entire premise fails to acknowledge the human element of the game AKA players and referee interference, in the idyllic but naive name of "punishing" the penalized team.
In the end, you'd either see games with very few penalties (and less intensity) as players tiptoe around each other, or total lopsided diving crapshoots where the referees lose control of the game.
It's not punishing a team, it's making the defense play hockey instead of whacking the puck. It's incentivizing teams/players/coaches to reduce the dangerous and non-hockey plays in the game. It's about improving scoring rates because the NHL is an entertainment product.
There's a bigger problem with cheapshots, obstruction, interference, etc. than there is with player fatigue. If they can survive the 3 on 3 gimmick with better players on the ice, they'll surely get by on the PK without icing.
Most teams carry depth PK specialists and everybody starts with a time out. Most of these consequences can be offset by coaching and roster adjustment. I don't think any rule change reduces intensity, maybe it's physical play that is a concern. As far as fatigue leading to more penalties, I'm not sure the numbers add up as much as perception does when it comes to play at the highest levels.
The diving issue has been partially addressed anyway and there are plenty of coaches that will bench diver if they they take themselves out of a play with that stuff anyway. I don't think diving would be any more of a concern than it already is.
Okay...so people are complaining and saying it would slow down the game too much...so how about THIS?
Okay so they change a couple things around.
1st.) On a PK, the icing line is moved back to the defending team's blue line. So they can ice the puck as long as they get past their blue line first.
2nd.) If a team DOES ice the puck from inside their defensive zone, they are unable to change players on the fly until the offensive team brings the puck back into the neutral zone.
So it will pretty much be the same as it is now except that teams can't change lines if they ice the puck from their defensive zone.
Edit 3/8/16: Changed my mind.