It's interesting that they are thinking of trying to complete the regular season as well. The push must come from NBC contracts, I would guess. But, at the same time, there are enough regular season games left to make it worthwhile for the players of the teams who are out of contention to come back and get into shape.
A little calendar work:
If a training camp can start on May 15, then games can start on June 1. The most games any one team has remaining is 14. Play every other day, and it's done in a month. So, about June 29. There are many logistical complications to this, not the least of which is that there are still a few interconference games left on the schedule. So, if the games are played in Edmonton, St Paul and 2 eastern cities, then there will have to be flights back and forth between them all. You could still have some awkward back-to-back situations there.
There are 189 games left to be played. One idea was 3 games per day in each rink. I don't think that's entirely possible for reasons having to do with which games are on everyone's schedule. But, let's say they can average 2 games a day in each rink. That's 8 games a day. Or, 24 days to finish the regular season schedule. Ending June 25....
Playoffs start June 27. This is much easier done, since the entire first 2 rounds will all be played in the same rink. No scheduling conflicts, and you can do every other day for everyone. 4 rounds takes at the most 60 days (including a single day pause between rounds so no one an advantageous back to back). Done August 24 or so. You could bump that up by skipping long rest days between rounds....eg, if Central Division first round series are finished in 6 games, round 2 starts 2 days early. Probably you can be done mid-August. Give the players a month break, and training camp for 20-21 starts mid-September?
That allows a full 4/7 games for all series, which, again, is ideal for NBC (at least in NHL's eyes).
If you shorten the series, then NHL has to give NBC a credit on next year, which they probably don't want to do.