When you get married you signed up to have sex with one person. One. Not two or three or four or whatever. You didn't sign up to have sex with anyone other than the woman you married.
But ... one is also not zero. You didn't sign up to have sex with anyone less than the woman you married.
There's a lot of discussion of libido mismatches online, and I totally agree with the point that no one has a right to demand that anyone else sleep with them. Absolutely true. No is no, and anyone, married or not, has the right to say no to anyone anytime. But I don't think it ends there. It is also wrong -- probably not equally wrong, but still most definitely wrong -- to think that you have the right to deny someone else their sexual needs.
This is sometimes taken aggressively, the old "if you don't sleep with me I'll find someone else who will" threat. I don't think it should or needs to be that way.
Simply put: You have these normal and reasonable needs. You are married. Your wife has the right to expect you will not stray and break your wedding vows. But I think that someone who refuses to acknowledge and reasonably meet their spouses needs is also breaking wedding vows. I don't know what specific vows you used, but I have a hard time seeing how that lives up to the whole "love, honor and cherish" thing.
So, my own opinion is this. You have needs that your wife is supposed to be meeting as part of being married to you. She has a right to refuse to have a sex life, but she does not have the right to make you not have a sex life. So, if she really cares about your marriage, she needs to be willing to fix this with you. There are options.
For one, there's the whole open relationship thing. Not necessarily completely open, do whatever you want with whoever, but an acceptance that if she's not going to met your needs you still have those needs to be met. It is not unusual couples to be in a similar situation with the roles reversed ... the wife wants sex but the husband is not interested. Finding a friend with benefits, with both spouses accepting it, doesn't exactly fit the traditional ideal of marriage, but marriage is more than just monogamy. A relationship of that sort could be a benefit to both marriages.
For another, there's, shall we say, stuff that you can do that does not require libido, it just requires caring about the other person and recognizing that they have one. I won't be explicit but I think you know what I'm talking about. If your wife no longer is interested in sex for her own desires, there's plenty of stuff she can still do. I have some personal experience with this myself; right before I went on TRT I lost my entire libido and had almost total ED. I didn't know what was going on with me, and I assumed it was permanent. And I devoted myself to the fine art of getting a woman off without using ... well, you get my drift. When one partner loses libido, I don't think they retain the right to expect fidelity unless they are willing to do everything they reasonably can to meet the other's needs.
Another possibility is that maybe her libido is fixable after all, and you're just not approaching her in the right way to get her to do something about it. Maybe her HRT needs refinement and she just needs the motivation to seek that refinement. Or maybe the problem isn't hormones after all. All those "tried and true methods" ... maybe what she needs to motivate her is something you've never thought of, or something that you think you are doing that's getting lost in translation. I used to bring my wife flowers all the time ... no particular reason, just because i love her ... which I thought made me quite the romantic catch. Turns out, my wife really doesn't like flowers. She's weird that way. But there are other things, mundane things that would never have occurred to me as romantic gestures, that really mean the world to her. Go figure. Maybe you just need to get her to open up to you about what it is she needs to make her feel the romance again; it might be something that never would have occurred to you. Maybe you can find this by talking directly, maybe some couples counseling is required.
The last option is the hardest, but it probably needs to be on the table, because it may actually be the best option, or maybe just because she needs to know that it's on the table so that she isn't taking anything for granted and blowing off your needs. And that option of course is divorce. Till death do us part is the ideal, but you don't get there unless you both are willing to work to get there. If she simply will not consider any reasonable alternative, if she thinks "just go beat off in the shower" is an acceptable answer that she can get away with because you won't walk, that puts you in a weak position.
The one option I would strongly advise against is cheating. That story almost never has a happy ending for anybody. Sounds like you already know that. If you are going to do something to end the marriage, just end the marriage first and then do it. It's so much cleaner and easier for everybody.