The problem with the Spelunky criticism is that it doesn't address that the damsel mechanic was put in the game purely for its value as gameplay.
The game is about navigating hazardous environments for treasure so Yu thought up challenge that involves successfully taking a destructible item to the end of the level. The item he would select to use would be based on two factors: its ability to visually represent when it was damaged and the challenge failed and to maximize player investment in the item so they will want to get it to the end.
He'd already used treasure everywhere in the game so choosing another, more valuable treasure, would have been redundant. A human NPC is kinda the perfect choice. Remember that Spelunky is about a set of universal rules that everything in the game adheres to. He had already established the rules for player damage so he wouldn't need to come up with an entirely new concept.
That he worked to give players three choices for the sprite is actually quite generous since he was trying to account for a variety of players just as he did by providing several Spelunker models.
Jesus Christ holy shit somebody else in the world gets it! Couldn't've put it better myself.
There is literally nothing wrong with having a mechanic in your game where a male character saves a female character, and especially nothing wrong if you allow the player to choose not only the gender of the player character, but also the gender of the to-be-rescued character. That last part is Yu going the extra mile; he was in no way obligated to do that, but he correctly realized that it should stave off any criticism of gender tropes.
Let's say I make a game about a carpenter, or maybe a plumber, who goes to a magical world of pipes and dinosaurs to rescue a princess. Is that me being sexist, having that as the core conceit of my game? It's definitely not progressive or imaginative, but is it bad?
Likewise, if I were to make a game about a woman falling in love with another woman, should it be critically lauded as the greatest thing ever, just because of its non-traditional relationship? What if I were to take the exact same game and change the relationship that of a "traditional" one? Would people be as all over that game as they would the other one?