I think a lot of it depends on 1) how well adapted your body is to exercise in a fasted state, and 2) how you run your S&S sessions.
1) The better adapted you are to operating in a fasted state, the less it matters
when you train. I've done my A+A sessions and aerobic work in the morning, afternoon, and evening. I typically run a 16/8 IF schedule, and have been doing that for 2-3 years now. I realize now that I probably could have saved a lot of time by doing away with lunch a while ago, and am doing just that now, but I digress. After the first year of IF (give or take) I never noticed much of a connection between
when I trained, how I felt, and what my weight did (to be fair, I've been within 5 pounds of a good weight for quite a while, so I don't even know what it would take to gain/lose a significant amount of weight).
2) S&S can be operated as a purist A+A program, a daily puke fest, or anything in between. To be fair, I think "somewhere in between, with a slant towards A+A," is the intent, but people have gotten good results using plenty of different work/rest strategies. In my experience, biasing your S&S practice towards an A+A style approach (start each set pretty fresh to minimize cumulative fatigue) tends to be most agreeable with a fasted body. Don't get me wrong, I'm sure some people could push it hard each day while fasted, but whenever I tried that it always left me feeling crummy.
There are, indeed, mixed messages about post workout fueling. I stopped doing it a while (2, 3, maybe 4 years ago?), and didn't notice any difference in muscle mass. I am going to make two bold statements:
1) A competitive strength athlete who is trying to maintain the heaviest body weight they possibly can while training multiple times per day should definitely take in some post-workout fuel to prevent muscle wasting.
2) An office worker who uses a few 15 minute strength sessions per week and lots of walking to increase their longevity and quality of life would get absolutely no benefit whatsoever from post-workout fueling.
Which end of the spectrum are you on? Personally, I'm at a weight that my body "likes" to be at, and my training is limited to maybe 3-4 hours per week due to the fact that I have a family and professional ambitions. Therefore, I feel no need to sync my eating and my training, and as far as I can tell, that hasn't caused any issues.
I've heard some people say delaying food intake after fasted exercise can result in an especially high surge in growth hormone once you do break your fast, but I honestly have no idea if that's true or not.
I say go for it. See what happens. You won't die