One of the things I loved in Dragon Age 2 was that one of the NPCs in the game will try to initiate a romance with you, instead of waiting for the player to try and initiate a romance with them. Now, Bioware got a lot of crap for this particular play. For one thing, the NPC was a man and would try to initiate romance with both male and female PCs. For another, Dragonage uses a point system to track how much people like you, and there's no way to rebuff someone romantically without losing points - there are ways to do it while losing less points, and you can make those points up, but the combination of being confronted by a potentially undesired suitor (regardless of gender) and having no way to say no without losing points felt like entrapment/blackmail by a lot of players.
I, however, loved it. Why? Well, for a number of reasons but most notably: your playing a character who is powerful, assumed to be decent looking (though some specifically make hideous characters for fun), important, and charismatic. They are a person going from rags to riches. They are a hero of a story that is bigger than most characters. Is it really so surprising that someone would be interested in them? Is it really so surprising that someone would make the first move and try to initiate a romance, instead of waiting and missing the chance?
This is something I try to bring into my games. Sex and sexuality is a part of who people are - whether they are an asexual virgin, a total hedonist who has sex frequently for no other reason than it is fun, or anywhere in between. It's not the only thing to characters, but it is there, and that means for both my PCs and NPCs have some thought to it. Beyond this, just what type of actions a character would find admirable, cool, or bad ass plays into this as well.
When I have these things in mind, and I look at the PCs in my game through that lens, there is often a lot to be desirable. For example, in my D&D game one of my players is playing a Tiefling Druid who has a different name and personality every day. Every long rest, the player rolls a D12 and gets a name off their list, and that name and core personality. The character has a 14 charisma, which means that - to me - they're able to carry their social anxiety as mysterious as opposed to creepy/being anti-social. This character is also a capable combatant. So when the party met a gnome Barbarian who calls himself Stab, he saw this person and found her appealing to his own ideals of what made someone interesting and attractive. She stood out, was a bit of an outcast among the 'other people', had something going on in her head that made people think she was weird, but was content to stand her ground and fight her fight.
And so this NPC made advances. They weren't overt. But it was clear to people that he liked the druid. And their budding relationship has been a fun thing to watch happen in the game, complete with the Paladin player laughingly ordering the druid player to "wife that gnome right now" after some of the things Stab did.
It doesn't always have to be romance though either. Your PCs have likely done grand deeds. If nothing else they've made money. It's not unusual for people to try to attach themselves to people like this - especially if they have a high charisma. So have NPCs be nice, be friends, and try to get in with the PCs even if for their own good.
It makes the world most real. It also shows that the world is reacting to the PCs. And when the PCs with high charisma are getting people flocking to them, and the PCs with low charisma aren't, it also shows how stats can work in the world passively to influence things around them.
Give it a shot. Don't push it too hard. But you may be surprised what happens when your NPCs start caring about who the PCs are. Sometimes the PCs start caring about the NPCs in return.