Thanks for all the feedback everyone! Now, to answer some of the questions and comments I remember reading...
We've been applying the death saving throw rules correctly, so it was never that that was affecting the danger.
While I play a dedicated healer when someone else DMs, that hasn't been the case recently, and wasn't an issue when the discussion was sparked. The party's healing capabilities come entirely from second wind, healing potions, and the few healing powers that the druid (multiclass shaman) has. Healing potions have been used more than anything else to revive unconscious allies, but limiting them, especially given the low cost of the weakest type, would be impossible to explain in-game and not my favorite solution out-of-game.
As for death saving throws continuing throughout the fight, what usually happens is that by the time the first PC goes down, monster resources are now low enough, and even lower when the PC is healed, that bringing them back down to 0 is hard to accomplish. In other words, no PC usually goes down until near the end of the fight.
So, often only having one fight per day seems to be the culprit that many of you have identified. It happens that way because we only have two three-hour sessions a month, so I tend to condense combat so a single adventure doesn't take months to resolve. For the same reason, we automatically level-up ever two sessions instead of counting experience points, because we'd like to level-up more than three-four times a year.
But, I'm definitely going to try only allowing the benefits of an extended rest at my discretion. By bringing back resource management and allowing me to vary encounter difficulty, the game should get closer to the baseline design, and likely involve slightly more risk. Mixed with, say, death at 2 failed death saving throws, and death at negative surge value, and this will certainly be closer to what my players are looking for, though perhaps too deadly.