See, I have actual experience with this particular issue.
All of these pretty much describe my experience in a nutshell.
Okay, so the story goes, in me and my friends' first D&D campaign, there were five players and our DM. One of my friends (a girl, who we shall call Grill) is actually dating our DM, and they've done some little flirty things in character before, mostly because Grill seems to like adding that dimension to her characters. And all of that is truly well and good, no one really minds.
But then came one particular arc when our party entered a brothel. Now, there were a lot of...particular women in this brothel, and there were various reactions from the men of our party to the situation. Now, for me, the romantic dimension isn't something I typically enjoy roleplaying or developing, and my character at the time was a very edge-tastic, mission-first kind of guy, so I roleplayed him as such. He had no reaction whatsoever to the advances of these women in this establishment.
Now, for my DM, who is one for realism, that decision didn't seem to sit well. He claimed that it wasn't a realistic reaction from any man, no matter his demeanor, and at the time I sort of agreed. I thought maybe giving my character that romantic dimension could help him feel more relatable.
Now, my character and Grill's by this time were pretty close friends, but they were nothing near a couple. The closest they had come was a few inside-shipping jokes. Grill and I had considered before maybe trying to make that work because it might be interesting, but we never had a good reason until now. So, with my DM's advice, we took the joke ship and made it a real thing.
So slowly, over the next few sessions, my character started developing feelings for Grill's, culminating in an extremely awkward first kiss scene (way too much to explain there, but it was very weird). After that, they were officially dating.
What followed was weeks of awkward character interactions. There were times where it seemed like it worked. But a lot of the time it really didn't. My character was way too closed off and unemotional, not interested in displaying affection, but Grill's was the exact opposite. Her character had a misconception about romance that made her think all there was to it was affectionate displays and dirty deeds.
There were some IRL reasons that it didn't work too. Now, surprisingly, the obvious one wasn't an issue. Despite the fact that Grill and our DM were dating, my DM had no issues with the roleplaying, he even encouraged it. Despite that, I still felt awkward having to do boyfriend things in character with Grill as her boyfriend, my best friend, watched. To top that all off, I am in a very good and steady relationship myself, and some part of me felt like I was betraying my actual girlfriend in some way. It's also extremely awkward to pretend to be dating one of your friends.
So there were a multitude of issues there, both in the game and out of it. I brought these issues up to Grill, and to my surprise she didn't really feel the same way. She kinda thought I was silly to feel awkward about the things I felt awkward about, and maybe it was a little, but the in-game reasons alone were enough for me to want to take my initial decision back.
And that's what I did, slowly. My character started to put more distance between him and Grill's character. Heck, it even culminated into some fights between them as their relationship broke down, her character mostly becoming unhappy with the lack of communication, mine expressing love, but not physical affection. To be honest, those scenes did lead to some cool moments as our party sort of fell apart for a bit trying to get us to work out our issues. But eventually, the decision was made. They would "take a break."
Eventually an NPC, Ben, joined our party, who Grill's character rebounded to, my character letting him be with her with no real fight. Sadly, it turned about that Ben was a traitor, and that actually turned out extremely interesting. At the end of the campaign, Grill's character and mine made-up and became good friends again, and she got back together with another well-established NPC, Dante, and they actually ended up getting married in the end.
And let me tell you, it was a good decision on everyone's part. Not only did it ease my conscience and let me focus on what really mattered for my character, it also led to some actual good story moments when it wasn't she and I who were together.
TL;DR Romance can be good, but only when it's done right. There needs to be a very good, in-game reason for it to be happening, it needs to actually lead to some good story moments, and it needs to be 100% consensual on both sides, be it player-to-player or player-to-DM. Make sure it makes sense for your character and that it's an aspect of him/her you want to explore.
Otherwise, don't do it. It will only bog down your character growth, or even the story as a whole, if your mind isn't set on it.