Yes, there are generally two camps on matters related to potty training. The "it's time" camp, and the "they will go when ready" camp. With my son, I was more in the "it's time" camp. He showed some readiness, so I encouraged him, and he got it pretty quickly and was trained around 30 months old. With my daughter, I never got to decide what camp I was in, because she was so interested and ready that she was trained by her 2nd birthday and it was a non-issue completely.
So, with that in mind, that I haven't personally dealt with exactly what you are going through, I offer my sincerest sympathy. It's a tough call. However, the biggest issue I am seeing is lack of consistency. I understand completely the reasons for it. (during the week it is busy---believe me, when my daughter was training, son was 4 1/2 and we were always on the go for one reason or another). However, busy or not, convenient or not, it still IS going to affect the training process. So, regardless of which camp you ultimately decide on, you should recognize that inconsistency is going to be a problem.
Yes, 3 year olds are rather limited in what they can control. Have you considered a middle of the road approach, whereby you give him some control? A common way to approach it is to eliminate the pull-ups (diapers) completely and give them the undies (which your son likes, correct?) all the time (daytime anyway). But, rather than putting him on the potty by a timer (every 30 minutes) maybe let him be the one in charge of that. He's 3 1/2. You can talk to him about this.
"Look, you are ready to wear underwear, but the pee/poop has to go in the potty and not in the undies. So, it is up to you to make it happen. I'll be glad to help whenever you need it. If you decide not to go in the potty, you will be responsible to clean up the mess."
(Maybe not quite so in your face, but you get the idea)
And follow through. Sure, you will have to help with the cleanup (for sanitary reasons), but he should be doing the bulk of it. And he won't like it. Usually kids decide pretty quickly at that point, that THEY prefer the mess going directly into the potty so they don't have to clean it up after. It is still their choice.
I would also look at the actual potty. Did/does he get a choice about what he uses? Both of my kids had potty chairs (handed-down wooden one made by an uncle of mine that my brothers and I trained on almost 50 years ago, and brand new plastic from Walmart), and neither of them used them more than a handful of times. They both really preferred using the actual toilet. But they preferred using it with an attachment on the toilet seat (so they felt less like they would fall in) and with a step stool so their feet could touch while they were seated. Every kid is different in what they will choose. So, here again is a way to let him have some control.
I wish you the best and lots of patience. Whether you give up for now and readdress it a few months from now, or change up your tactics and continue working toward success NOW, you will need patience.
ETA: Just wanted to throw in that when I was training my son, he was our first child and I didn't read tons of "how to potty train your child" information. So I didn't have a lot of confusing directions about how things were "supposed" to go. I just followed my instincts. I never used a timer with him. I was cognizant that it had "been awhile" since he had peed, for example, but I didn't have a timer going. I encouraged him to try shortly after a meal or a lot of milk/water had been consumed. I required he try before we left in the car or he took a nap, etc. But I never used a timer. Not sure if that is of any value to you or not. Sometimes, I think, if we can shut out all the "noise" of what everyone else thinks "ought" to happen, we can feel our way through things and listen to our kids and what might work best for them a little better. What works for my kid or my neighbor's kid, may not work for yours. But you are going to be more in tune with your kid, if you let yourself stop worrying and comparing to everyone else's. (not saying you are doing that, but it is easy to start 2nd guessing ourselves, ya know?).