"Why is the word "lady" seemingly prevalent in usage?"
Both those words; lady and gentlemen, were derived from the days of chivalry. Feminism killed chivalry with it's insistence that women did not have to measure up to ladylike behavior. As ladies went, so did gentlemen.
However, it is the behaviour, and not the term that was discarded. Most women like the idea of being -treated- like a lady. But they don't necessarily like the idea of always having to -act- like a lady. The modern day dilemma is that a lot of women think that since feminism has given them this choice, that they can turn it on or off and expect the intended results (lady-like respect by any man reverting to a gentleman on the spot)
But of course, men have never been given any such choice with their behaviours. So they don't understand this idea of a lady one minute and a tramp the next. Depending on the man, he treats the women he knows by their long term reputations rather than their insistence on how they feel at the moment. A lot of other men disrespect women they don't know for what they see as their flip-flop behaviours and unrealistic expectations on how men should treat them based on their demands of entitlement that feminism has provided them.
So the quick answer is that most women like the idea of being seen as a lady. Most men are much more pragmatic and don't care for being a gentleman, or even considered one, once the term lost it's respectable role, and became a badge of exploitation.
Feminism has changed society. And it continues to do so. We need to be careful what we ask for. You can not unfry an egg.