Shooting percentage does rely heavily on luck, but it's not entirely (or even mostly) luck-based.
A player like Ovechkin, that has a world-class shot, has a higher baseline shooting percentage than an average player. His career shooting percentage hovers right around 12.5%. During a "good luck" year, he'll have a shooting percentage near 15%. During a "bad luck" year, he'll have a shooting percentage near 10%.
Evander Kane is another player known for shot volume. Since the 2010-11 season, he is 5th in the league in shots taken by a forward. His baseline shooting percentage is 9%. During a "good luck" year, he gets close to 11%, during a "bad luck" year, he is right around 7.5%.
So yes, sometimes players get lucky, and sometimes they get unlucky. However, it's bonkers to think that shot quality doesn't matter at all and that the baseline is the same for all players. If Kane started shooting at 11% constantly, he'd be an All-Star and be talked about among the league's elite players. If Ovechkin started shooting at 11% constantly, people would be wondering what was wrong with him.
For the record, Jay Beagle's career baseline appears to be around 9.5%. Better than the league average over the last decade, but not Ovechkinesque.
https://www.hockey-refere...ce.com/leagues/stats.html
Shooting percentage is inverse save percentage. In the 8th percentile over the last decade, but up to 9.3% so far this season.
