I was my SO's carer for two years whilst he was hit with depression. I felt I was his mother rather than his partner.
The fact your wife wants to want sex is a good start. The fact she's trying (although with no great results) and still wants physical affection are all good signs.
The tablets can be an absolute kicker. When I went on SSRIs I ladieparts just went dead. Masturbating was like fingering a brick. There was just nothing there.
I think the first question has to be, Do you still want to work on this, or are you just looking for permission to have affairs/casual sex?
If you're happy to redouble your efforts and keep trying for a while, couple to things I'd suggest.
Couples therapy with a sexual angle. We ended up with a nice way to create mentally & physically safe space where we could just have sexy fun with no requirement for full on sex. It took the pressure off, which helped the natural desire come back on.
Get her medication reviewed. There are so many different types of treatments. My own life was much improved when me & GP opted to focus more on the pain and exhaustion, instead of the depression. I ditched the SSRIs & painkillers, and went over to Lyrica. The pain relief and energy increase diminished the depression anyway.
If you really just want to go the route of being life-companions and getting sex elsewhere. That's your choice... but please, please understand she'll find out. At best she'll realise straight away, at worst she'll suspect for months before she actually knows, and those months will be hell on earth for her. No matter how well you think you'll cover your tracks, she'll know. So if you want sex outside the relationship, and not cripple her with the pain of finding out, you're going to have to have that conversation with her first.