This is a very interesting discussion. We don't really have data to project what would happen if teams did it earlier, we can only predict
So yes the fact that teams require 2 units is a factor; 18 skaters on a team so you second unit is your middle third. Also while teams practice the goalie out play, they are limited by the number of players. 11 players on the ice at a time so the reality is the entire 21 skaters on the roster (including 3 which dont dress) would still be 1 short to have two offensive units and two defensive units. And consider there are players who would be playing both sides of the goalie out scenario in a game situation.
Now people aren't talking much about teams down by 2 or even 3 pulling the goalie with more than 2 minutes to go. They'd have to use 2 units there.
Also another factor that the essays written in the 1980's do not cover; the icing line change rule brought in since 2005; that makes 2 important changes. Defensive teams less likely to simply fire it down the ice blindly as they may have when they could get a change. This is also causing the team on defense to use time outs more often in these cases. I think it is also causing offensive teams to use their 2nd units more. It's a common strategy for coaches since that rule came in to get fresh legs out there when the opposition ices it in order to get an energy advantage (and i'm talking about the entire game here not just goalie out scenarios).
So alas here's the way I see it (all these scenarios i'm not counting less than 5 seconds to go in a period or delayed penalties):
6 on 5 the simple odds indicate the team with the empty net to shoot at has more of a chance at scoring than the one with the extra attacker. Maybe a good study someone can do is look up in the last playoff how many goals were scored in these scenarios.
here's a list of empty net goals by team last year
http://sportsillustrated.cnn.com/hockey/nhl/stats/2011/teams/team_scoring/0_byEMPTY_NET_GOALS.html
highest is 14, lowest is 4, median is 7. Again if you throw out <5 second or delayed penalty goalie out scenarios i'd gander the number of goals scored BY the teams with the empty net might be half. that.
.
Now of course the simple odds are that you're more likely to score a goal then get scored on with the goalie out so when you need a goal badly why not? Let's say you tell a coach they have 2 options;
leave the goalie in while losing 10 times in the last minute, you'll score once and get scored on once and have no goals the other 8 times.
OR pull the goalie, you'll score twice, get scored on 4 times and have no goals the other 4.
Well any rational coach would look at it as one extra tying goal is worth the 3 extra games losing by 2.
So why not do it earlier? The coach is more likely to get blamed. I'm sure we have all been in situations where a team is down by 1, pulls the goalie and gets scored on, puts the goalie back in and scores; so they lose by 1 with the empty netter being the difference. It would look bad on the coach if this happens due to him pulling the goalie too early. I would also gander that you put both teams second units out there (and some teams do use 2 units defensively in that situation) and the defending team has more of an advantage as scoring a goal requires more precision than defending
Now when it's 6 on 4 I think teams should get more aggressive. Being one man short you can at least take away all the passing lanes (4 on 5 or 5 on 6) but being 2 men short, there really isnt a viable strategy other than force it away from the slot. The icing factor does help somewhat but it isn't a good enough trade off; they have to get the puck, have a lane to clear it and get there first. I've only seen this sort of thing experimented with once; around 1998 Mike Keenan was coaching Vancouver against toronto. He was down by 1 or 2 and 3 separate times in the 3rd period pulled the goalie when on the powerplay. The first two games he put the goalie back in on the fly when the penalty ended. Third time he got scored on.
Now a rarity; 6 on 3. Despite what they say, referees are less likely to call a penalty on a team already short handed and less likely to call a penalty late in a close game so combine those factors and you'd be lucky to see a team finish a one goal game 2 men short once every 100 games. But as brought up, what would the natural odds be? Would it be beneficial for a team with a 2 man advantage in a tie game scenario to do it? Well probably not with a loser point; if its in the third, they wouldnt be wise to gamble away that 1 point for 2 when they still have OT and a shootout to get that second point. If its in OT, they'd lose their point if scored on with the goalie out. But if your question is simply which team is more likely to score a goal in a 3 on 6 scenario? I kind of think it would be the team with 6. There's no possible way to defend that without hoping for dumb luck. Think of all the different plays and strategies the offensive team can design for that? If they put 4 men deep and 2 on the point, the defenders would have to choose between either covering 1 point man and leaving 2 men alone in the slot or leaving 1 man open in the slot and both point men. If I was coaching and losing in the 3rd at any point I would probably pull my goalie with a 2 man advantage.