I don't think Swayman is greedy for trying to get what he thinks he is worth. Goalies are underrated and underpaid in todays NHL as is.
More importantly, I don't know why fans side with the team against players in situations like this, it's not like Swayman is a third string goalie trying to rob the team. He's their star starting goalie and the main reason they won a playoff round. He has the potential (if he isnt already there) to be a top 5 goalie in the league. If the team hopes to remain good with their aging core, Swayman is going to be their biggest piece to do that.
At the end of the day, teams want to pay their players as little as possible, and players want to make as much as possible, it's just the nature of the business. That being said though, players are the ones who bring all of the value to hockey and to their team. People come to watch the games because of the players and the play they bring to their team.
Also, sure, you can say that a player should take a discount to play for your team, but on what ground should they? The team trashed Swayman during arbitration, they haven't won much in the playoffs during his time, I don't imagine there is very much goodwill there. The Bruins traded away Ullmark and made Swayman the future of their goaltending before signing him to an extension, giving him all of the leverage. What is he supposed to do, take a pay cut despite the writing on the wall being that he's going to be their most important player? It's not the players job to build the team, its the teams job to anticipate what their players are worth, will be worth, and build the team accordingly. Players don't owe the team discounts, and even if you think they do, that's because the team and the NHL have successfully tried to build the narrative that players have to take paycuts to win, it's a narrative that only benefits the teams and owners and harms the players.
And keep in mind, a lot of the time when people point to players that have given discounts, like Hedman, Kucherov, Stamkos, etc, they play in markets that don't have income tax, or in the case of a player like Crosby, have decades of history and goodwill with the team. If Hedman played in LA, for example, he'd be signed to north of $11M a season just like Doughty has. If Kucherov played in a different market he'd probably be making north of $12M. It's not always a fair comparison.