Define big. Define hugely influenced. This is the root of the problem, not that luck is or isn't a factor. I doubt anyone disbelieves the presence of luck in sports. It's in part what makes it so unpredictable and entertaining. But analytics types often make claims in a way adversarial to skeptics serving little purpose beyond showing their zeal. I get that using exclamation points can perhaps shift the goal posts and the spectrum of the debate but it doesn't serve the merits of their findings much. I'm a fan of the intent behind the field and by no means made any sort of blanket statement if you'll read what I wrote again. If there's a particular study with results indicating an exact measurement of luck (beyond PDO) I'm all ears. Again, what I take issue with are some of the hasty conclusions drawn from one-dimensional metrics and cartoonish wording often used for effect.
http://hockeyviz.com/txt/oscar
http://hockeyviz.com/txt/cordelia
Cordelia predicted that the favorites and overwhelming regular season champions only had a 14% chance of winning the Stanley Cup in 2016-17 and the older version (Oscar) predicted that the favorites would only have a 18% chance of winning in 2015-16.
Other models and their probabilities (I'm on mobile so I can't go through them right now):
When the best team in the NHL has an ~83% chance of not winning the Cup according to an average of a variety of different models it's tough to deny the existence of a significant luck component. Contrast that with the NBA where most models gave the Warriors a > 50% chance of winning the championship (one example:
https://projects.fivethirtyeight.com/2017-nba-predictions/) and it gets really hard to consider hockey the same as every other sport. Betting lines similarly reflect the uncertainty in outcomes. These models aren't perfect but I would guess that they are closer to reality than most people believe.
Luck isn't just "bad bounces", it's things that happen that aren't really intentional or under one's control. It's nice to say that "more traffic = more goals", but that's only true in probabilistic terms. You still have to rely on the shot getting through and getting a favorable bounce, or finding a hole in the goalie rather than bouncing harmlessly off his pad, etc. Those individual things aren't reasonably under one's control, even if the general "make traffic" idea is.
The Ovechkin Era in the playoffs has to this point certainly been a failure. Full stop. Without question it has been on a team level given their regular season success and expectation levels. There are no mitigating circumstances the organization should believe that weaken their resolve in doing what it takes to improve. That was the root of this topic from the outset...how they continue to appear to have made very little sense out of what happened and, by extension, form a coherent way forward beyond showing up to do it again. Outside influences certainly do exist but it serves no purpose to give it a second thought if they have the mental toughness required to understand what's within their control and do their best in improving in every area within that category.
From a championships point of view absolutely they have been a failure. This offseason has been a failure and why I am in favor of at the very least a new head coach (for reasons you have pointed out well), and why I wouldn't be opposed to a complete house-cleaning given that they don't really seem to know what they are doing in terms of re-tooling or re-building.
But the fact that they even built those expectations to begin with and been a regular season powerhouse is a resounding success IMO. It feels like more of an emotional letdown when they don't follow through with postseason success but it makes for a better and more entertaining ride. How many teams can legitimately make the claim that they have built a consistently entertaining ride for close to a decade? Honestly I feel bad for fans who think that the only measure of success is a championship. Kings fans right now aren't having a very good time right now, for instance. Having a championship is nice to look back on but ultimately doesn't mean much soon after it's won IMO. I couldn't deal with 10 years of Hunter hockey even if somehow they won a Cup during that time-period. Perhaps it's just that I am entertained in different ways by sport and I define success differently.
Again, this doesn't mean that I think changes shouldn't be made this offseason because moreso than any time since the Oates era it seems like no one has any idea what they are doing and the team appears to be heading in a significantly worse direction than they have to.
But can you quantify that it is indeed closer to the truth? Aren't you essentially speculating just the same? Cup champs also tend to be able to do three things that pump their PDO: they manage to manufacture and convert high quality scoring opportunities 5-on-5 in order to fuel Sh%, they're better able to dictate shot quality 5-on-5 and get strong goaltending. So there's a sensible explanation for what you chalk up predominantly to luck, particularly over 24 games or so. Much of what makes a strong playoff team is the very ability to find and sustain a level of remarkably exceptional level of play and that's not just shown in the number of bounces that go their way but also in timing, cohesiveness and decision-making. It's all needed to win and, whether results-based or not, something these Washington Capitals have never gotten to. As I've said, it's not always the numbers per se but subtle conclusions drawn from them. Even PDO only accounts for 5-on-5 play so it's hard to point to that as the ultimate indicator of who is and isn't lucky.
Peaks and valleys in PDOs aren't repeatable in the slightest. Corsica has a stat called xPDO (expected PDO) which IIRC incorporates shot/shooter-quality and goaltending quality rather than just assuming all shots are the same (unfortunately Corsica is down right now). I'll double-check when the site is back up but I am almost certain that every Cup champ since 2008 (when the database starts) has exceeded their xPDO.
Yeah you can catch fire and have hot and cold streaks but no team or even player has been able to show a reasonable level of being able to "turn it on" beyond what can be explained by simple variation. Pittsburgh is a nice example of a team able to create high quality chances but so is Chicago and how did that turn out for them this postseason? They hit a dry spell despite icing one of their most talented teams ever. It's hard to make the claim that Chicago suddenly didn't have the ability to turn it on given that they have shown that ability in 3 prior postseasons (and another deep run) in recent memory. It's more likely that these streaks of hot and cold play are simply things that happen and that are unreasonable to expect any team to control (even if they are theoretically possible). Fleury is a perfect example of this. He's not nearly as bad of a goaltender as his prior postseasons indicate but he's also not nearly as good as he showed this postseason. Did he suddenly just willfully turn it on? I find that hard to believe.