No, players contracts are not based on 82 game season excluding playoffs. Their contract salary $ represents a slice of 50% of hockey-related revenue, including the revenue pulled in from playoffs. From each paycheck, money is withheld and put into an escrow fund, and if for some reason the playoffs are cancelled, such as in COVID, they don't get any of that escrow money back. In the case of COVID, there are knock-on effects because last year players were paid money they didn't end up earning, because there were no fans in the seats during playoffs. So the players will collectively be paying money back to the league out of their paychecks for the next ~5 years, all because one playoffs of fans in the seats were missed. When you look at it this way, players are being paid for the revenue earned in the playoffs, before the money from those playoffs are even earned!
In normal circumstances with fans in the stadiums, if there were less NHL games played, let's say a lock-shortened season, then all players will make less money on account of the league making less revenue. That's not because they didn't play the games, it's because there was less revenue. If the league, for example, fully compensated for the lost gate revenue by putting ads on jerseys, then players would get paid their regular salary, despite having only played about half the games.
If in the 2022 season the league puts ads on jerseys and it doubles the revenue unexpectedly, then a player who was on a $5m/yr contract will start making $10m/yr (well cap will go up, new player salaries will inflate which will take away from the doubling revenue pool, but you catch the drift).
It's not that players aren't paid for playoffs -- all players are. Players that don't play in the playoffs get paid for the playoffs, even if their team didn't qualify.