But it's not clear that pain aversion (or sexual desire) can be fit into the same narrative.
I don't see why not? I feel like you already stated in an earlier paragraph how it can be fit into the same narrative, when you said:
What of the third example? Well, that depends on what the manager feels about their own sexual urges. They may be more or less endorsed, depending on various factors of the manager's values and social circumstances.
Nothing says that higher-order preferences would have to be as simple as "unconditionally endorse" or "unconditionally unendorse": you could have a complex higher-order preference that endorsed the lower-order one in some situations and unendorsed it in others. That seems like a very straightforward extension of the framework, and certainly much simpler than saying that we should abandon it entirely.
For the record, I do suspect that the model of lower- and higher-order desires is too simplistic, and that people who feel that higher-order desires should always win may be identifying too much with the rider; my model would be something more like "there are different shards of desire, each seeking to achieve its own interests and their power balances shifting depending on different situations, with things being closer to a scheming court full of constantly changing alliances than to any hierarchic system". But I don't feel like your examples are particularly compelling demonstrations of the model being flawed.
You're doing a little slight of hand by throwing all "avoiding pain" in one large bucket (and then deciding that you want to keep some of it), but then instead of analyzing "avoiding fear" as applied to one specific threat (and then deciding that it's "irrational"). You could just as easily say "no, I don't approve of feeling pain when I need to kick an important game winning goal. It's high order vs low order", or "I absolutely approve of keeping my fears, because they also protect me from real threats".
I don't find the distinction to be useful, except in modeling how other people relate to their own impulses. Even when people tell me that their fear is "irrational" and that they want it thrown out, I treat it more like the way you refer to the pain aversion case, and it works.
For example, my friend was telling me about her "irrational fear of heights" that she wanted gone, so I had her climb up a rock wall over concrete and had her try to hold the frame that there was zero risk and that the fear was entirely irrational while I kept pointing at all the failure modes and asking her to explain how she knew that wouldn't happen. This forced her to take the fear seriously, and once she did she was able to integrate it into her decision making process more efficiently and therefore able to be less paralyzed by fear when rock climbing and without throwing out any of the valuable information that the fear has. Similarly, there are times when you can look at the pain and decide that it's not necessary and watch it melt away into ticklish sensations or nothingness (and then kick the ball without wincing or anticipating badness).
In both cases, I'd look at it as a signal that there's value unaccounted for in the decision you're wanting to make, and once you properly account for it, all conflict and discomfort vanishes.
Suppose a footballer with a broken leg is steeling themselves to kick a penalty despite the pain. Suppose a statistics lecturer is about to take a plane to a conference, and is trying to overcome their fear of flying. Or suppose a manager is meeting a client, and is trying to overcome their sexual attraction and stay professional.
All three situations can be seen as a conflict between higher order rationality and lower order urges, but they are actually different. In every case, there is an instrumental reason to overcome the "urge", but the general attitude towards that urge is different.
For the footballer, they want to overcome their current pain avoidance instincts, but they don't want to get rid of those instincts entirely. They generally want to avoid pain, but, for the moment, they have something more important to do - win a game - and would want to put their pain aversion preference aside for the moment.
In contrast the statistician with a fear of flying wouldn't mind having that fear expunged for ever. They know that normal flying involves no large risks, so a fear of flying is wholly irrational.
Thus we see pain aversion as an endorsed preference; we wouldn't want to get rid of it. Fear of flying is an unendorsed preference: we would toss it out if we could.
What of the third example? Well, that depends on what the manager feels about their own sexual urges. They may be more or less endorsed, depending on various factors of the manager's values and social circumstances.
For fear of flying, we can fit that fear into a simple narrative of lower order preferences being overruled - or not - by higher order rationality.
But it's not clear that pain aversion (or sexual desire) can be fit into the same narrative. Since we don't want to override our pain aversion, it might be fitting to see pain aversion as higher order preference itself - or at least "not overriding pain aversion" might be a higher order preference, while pain aversion is a lower order one.
One can complicate the picture further by considering more edge cases, but the simplest interpretation is that the simple "higher order versus lower order" does not fit pain aversion or (often) sexual desire.