A few comments...
While I agree that sexual intimacy is a subset of intimacy in general, I don't think "an already-existing friendship, where trust and integrity already exist within the marriage, and where you genuinely enjoy one another, i.e., where the framework is already in place" is necessary for sexual intimacy. I can be sexually intimate with a ****buddy, prostitute or my right hand where those factors don't come into play. But, for the sake of arguments, sexual intimacy with those factors involved is much better than without.
Defining sexual intimacy is important. I believe OP has spoken true at least regarding his definition of sexual intimacy. Depending on how you look at sexual intimacy, you can have sex without being sexually intimate, just as you can have a conversation about the weather without being emotionally intimate, or attend church with someone without being spiritually intimate. Regardless of the type of intimacy, the word "intimate" implies a certain level of connectedness, openness and vulnerability. You can bang a prostitute without any kind of connectedness or vulnerability. Casual sex is, by its very definition, lacking in connection and, therefore intimacy, even if genitals are coming into contact.
If OP just wanted to get his rocks off without intimacy, he could bang his wife, ejaculate, and call it a day. But he wants more than just sex, he wants genuine sexual intimacy.
"Letting herself go" could use some clarification. Yes, she's gained weight. Do you know the how and the why of this? There are two major components here, either or both of which may be driving this. They are exercise and diet. To the first, is your wife not motivated to be active? I'm not talking about training for a marathon here, just spending some time each day not on the couch or behind a desk. There are many reasons a woman may lose motivation to move, both physical and mental. Isolating the underlying cause may be helpful. Similarly with diet, does she deliberately eat poorly or too much. Again, there are most likely underlying issues here. It's easy to use food for comfort, and when we do, it's not gonna' be kale and carrots, it's gonna' be ice cream and cheesecake.
The weight gain is most likely merely a symptom rather than the disease itself. Your age only supports this theory further. By mid forties, it gets much harder for a woman to maintain weight, and it can be easy to just give up. Moreover, women can also have a variation of a midlife crisis as they exit childbearing years, even if they have no desire for children. Thinking she is not as desirable as when she was young can be demotivating as well. Some just lose any instinct for self care.
You say you didn't separate over infidelity, but you don't say why you did separate. Did you separate because you were already suffering a sexless marriage or did that come after? Was she gaining weight before the separation or after? If it was after, the separation may be emotionally troubling to her, even if you separated amicably. Lots going on here--to many unknowns to pin anything down yet.