Stories for many, many years made frequent use of the damsel in distress trope, where female characters are depicted as needing men to save them in big, dramatic displays. People eventually realized this was very sexist, which lead to an increase in the depiction of “strong, independent women” characters.
That said, although sexist the damsel in distress trope did show a useful narrative purpose: it allowed the (male) character to demonstrate how much they care about the female through some feat of heroism. As in “I love you so much I’d fight a dragon for you”. Additionally, while strong-willed female characters who have their own character arcs not dependent on male characters are good, when executed poorly results in a character who exhibits no vulnerability to make them sympathetic and grounded, and seemingly does not require assistance from others physically, emotionally, or socially (and to be fair, any human being is going to have moments of weakness where they need help from their friends and family), which makes them come off as a sociopath. It also has a tendency to make male characters appear bumbling and ineffectual.
This also ties into reader expectations. Male readers like to believe they’d be the type to drop everything and save the princess, whereas women like to believe they’d be capable enough to not need rescuing in the first place. Neither men nor women like to see themselves as incompetent. This leads to a paradox. If a female character is unable to save herself she is seen as weak and dependent on men to succeed. If the male character does not save her it makes him look incompetent and implies he doesn't care enough about her to rescue her. Men and women also like (both vicariously in fiction and in real life) partners that come off as competent.
I have two characters who are in a relationship. The male is a very chivalrous type who values his partner deeply. If she was ever in danger he would drop everything and perform whatever heroic feat necessary to save her (say, Bleach’s “storming Soul Society/Hueco Mundo level” feat). The female, on the other hand, is very self-reliant and does not like to place her agency in the hands of others.
Right now, the male character kind of comes off as ineffectual because while he cares enough that he would do anything to help his partner (and that's supposed to be a virtue of his), she isn't the type to allow herself to be put in that position in the first place. She comes from an abusive background that results in her being scrappy and trying to avoid depending on others, but because of this the idea that someone would value her enough to risk their life to save her would be immensely touching. I've tried engineering situations where the man is able to show how much he cares by trying to save her life (this is a story with a lot of action), but nothing feels right. Even when I tried the old standby of "love interest gets kidnapped and partner rescues them" it made more sense given her skillset that the female would escape before the male ever shows up, making him look like an idiot (long story short, he's basically the paladin to her thief). The male character is very well suited to making big displays to show his affection due to his character (he's very socially awkward but when put under pressure he acts very heroically).
I've tried making them a battle couple (which they normally are), but it doesn't quite work. The statement usually made in such a gesture is "I love you so much that when you are vulnerable I can and will protect you". Which requires at least momentary vulnerability on the one character's part and competence on the other's. I've tried making the male support the female emotionally, but due to the plot she ends up going to other people to find emotional support. There are also plenty of scenes in which the female saves the male physically (and the same message is present there), but writing the opposite is hard. Both of them are supposed to come off as competent and caring about one another.
Given this, how to I deal with the paradox of allowing a male character to save a female without making the female character seem incompetent, or vice versa?