These situations are never just about the lack of sex. They are about the lack of desire, admiration, validation as a sexual being and concern for a partner's happiness.
This is why any posts that suggest the OP would be within his rights to seek sex elsewhere are missing the point and the subtlety of the dilemma. The OP hasn't once expressed a desire to seek sex outside his marriage. It's patently obvious he loves his wife and wants sex with her. Sex for him (like a lot of people) is probably not a discrete activity where it is possible to keep his emotions separate. If he were to have sex outside of his marriage, even if that was an openly conducted activity with the full knowledge of his wife, he might still run the risk of losing his marriage if he developed feelings for the other woman and experienced a pull away from his wife. Not everyone is able to divorce feelings from sex and cannot view it as merely a means to sate a biological urge. A secret affair might be equally unappealing, because the OP might again be like many people who don't want to become deceivers and liars in their quest for sexual satisfaction. His personal integrity might be more important to him than sex.
If any of that is the case for the OP, then the best option is to change the conversation. He says that talking produces temporary change and never seems to deal with the root-cause of his wife's lower libido. When there is a script like that running on repeat in a relationship, it makes no sense to re-enact it because it will never achieve a different result.
Better to get a third party like a psycho-sexual counsellor to supervise the conversation and make appropriate interventions. What often emerges in those sessions are unhelpful 'sex associations' that were created in childhood or after early sexual experiences. These too run like a silent but deadly script during later sexual relationships.
It's true that some people are asexual and it can't be ruled out, but a longstanding aversion to sex usually has its roots in past or present relationships. This can include the way adults were socialised as children about sex by parents and care-givers, damaging early sexual experiences (not just abuse), unhelpful beliefs about differences between men and women's sexual appetites and sometimes just sexual incompatibility with a current partner.
So I'd change tack OP and be as honest as you can about your unhappiness. You are not wrong to feel this unhappiness and dissatisfaction and you should feel no sense of shame for wanting to have more sex with your wife. I would frame this in terms of your unhappiness and the other negative effects this is having on your wellbeing, so that there can be no misunderstanding that this is 'just' about sex. But make this a prelude to a request to go for some therapy as a couple.
If your wife refuses to go, isn't honest with you about the real problem or makes empty promises in order to stop the discussion, you've got some tough decisions to make. I think you'll feel better about yourself if those decisions are ethical and won't compromise your personal integrity. Good luck.