TBH when I saw the bit about boarding school at a young age I did think OH GOD NO. I mean feel free to put me right here, but even the nicest of them aren't really places where kids can safely be vulnerable, cry about stuff, go be cuddled by Mum and Dad, have pets to be devoted to, have grandparents or aunties to pop round the corner to, they're more like a junior version of working away on a business trip for long periods of time.
You say you're having individual therapy and thank heavens really, but how are you when it comes to being intimate and vulnerable with people, cherishing and nurturing them? Because that's what parents have to do.
Read up about the ghastly experiment where baby monkeys had to choose between wood-monkey-mum-with-milk and cuddly-fake-monkey-mum-without-milk - they went for the cuddles. The #1 thing that makes parents parents is that they're the one the kid goes to howling for a kiss and a cuddle when they've fallen over. You want kids - to be a good Dad you're going to have to do that.
You say you show love in practical ways and being willing to get stuck in is certainly a positive trait. Nobody is saying stop that :D
But do you think your wife also had a regimented childhood and married you because at the time you were as, um, emotionally limited as she was?
I'm probably projecting like crazy, to be frank, but my Dad had a boarding school-and-wicked-uncles loveless childhood so he internalised that good behaviour and "doing well in life" and "fitting in" were important and he literally, quite literally, had no idea what a loving family life was like. Not his fault, he'd never seen it done. He ..sort of...loved his dogs because his housemaster had dogs so he had seen how that worked!
My Mum left him when she realized that he didn't really understand that there was more to marriage than paying the mortgage and not sleeping around, and he then married someone who'd had a no-good type for her first husband, and so presumably valued his stability, though given that they then bickered (in JUST the way you describe in the 'glass' incident) FOR THE FIFTY YEARS OF THEIR MARRIAGE does make me wonder - well - just why? - it seems rude to ask and I'm sure if I did I would just get "I don't know what you mean".
We normally see the "one partner changes and the other doesn't" problem with weight loss, fitness, drinking, or work & finances, but emotional growth is certainly a thing where couples can grow apart too.
Here's something to try. If you were watching your life together with the sound turned off... what do you see?
Language is a relatively modern arrival in our mammalian evolution and we're quite good at saying things that aren't entirely true, but we are mostly still crap at disguising our facial expressions and body language. She may say XYZ, you may say ABC but in silent movie mode what do things look like?
I'm not saying be a dick to her. She's a fellow human being, however things play out, and deserves consideration as we all do. But just live your best life (oh and just get on and book the joint counselling and be very willing to listen to her and think about what you hear) and whether the marriage survives or you end up as friends (basically it sounds like you're slightly grumpy housemates right now) or as ex-acquaintances ... will come out in the wash as a side effect of living that good life. Make friends. Make plans. Go places. Do stuff. Think about what you'd do if you had just 10 years to live.
Oh and if you make friends with happy relationships - get them round for dinner if you can - there's nothing like a good role model for inspiring people to change.
And if you never had a pet - you are a grown up ... you can have one now if you like - even something like a hamster can teach you a huge amount about caring, life, love, sickness and old age, stuff that will stand you in good stead with your human relationships. Just a thought.
Anyway good luck OP, you're doing the right thing trying to break out of the current pattern for a start.