Personally I don't think that the white knight situation is necessarily a bad one. It is, however, an overused one and does seriously lack proof of a meaningful relationship, the care and affection the two characters are supposed to feel for one another largely meant to be taken for granted. I think there is romance in the idea that someone's willing to risk their lives for another, it just needs to be better established how much the two characters mean to one another.
Yahtzee is right that all too frequently relationships between a protagonist and a love interest is either shown in its immediate beginning or at a sudden and tragic (though seldom surprising, at least to the player) end. Say what you like about the Mummy series of movies with Brendan Frasier; at least it shows how his relationship with Evillan stays stable, they marry and have a son who's grows up into an intelligent and emotionally stable kid.
It's that sort of relationship I think games could stand to develop. Rather than -end- with the protagonist and their love interest acknowledging their feelings for one another, have the romance develop earlier on. Rather than have a sudden, instant oh-I'm-so-in-love-with-you moment right at the end, have a BUDDING romance at the beginning. The protagonist could be supported by his love interest or could be the one doing the supporting. They could even trade on and off in a genuinely balanced relationship, helping each other fight their battles, in some cases perhaps literally.
As much as some folks may not like the ideas of sequels, it would be a useful plot device for them. If two characters are shown having a long-term relationship, its evolution can become an incentive to play the next game in a series. Perhaps ending with one character contemplating if they should propose marriage in the first game, then partway through game two, they actually pop the question.
Video game character's relationships could definitely stand to gain a bit more maturity and complexity. I don't think the whole white knight situation should be thrown aside entirely...but maybe we could stand to start exploring what happens AFTER they ride off into the sunset.