You've gotten a lot of seemingly conflicting advice . . . Let me add my 2 cents' worth.
Our older son (2nd child) is now almost 30, and he's a very sensitive guy. He was a 'sore loser', also and would also cheat to win (a BIG NO-NO in our family). I took his sensitivities as rebellion when he was a child and punished WAY too much, though.
Yes, do punish him for bad behavior. But be sure to praise and reward him for good behavior!
Yes, do affirm his feelings, but be sure to let him know that everyone ELSE has feelings, also, and it's NOT 'all about him'.
Yes, this is about your parenting, but that means it's also within your power to turn it around!!
I'm guessing that this is almost an overly 'big deal' to you and your husband, and your son is playing into that. I agree that the less attention he gets for his meltdowns -- and the more separation he experiences from participation, the better. I also agree not to talk to him DURING the meltdown, but discuss it with him at a calm time and with diplomacy.
You're good parents, it's just that none of us got a 'one-size-fits-all manual' for parenting. I've heard it said that there are as many ways of parenting as there are parents. I'll go a step further and say that there are as many different ways to parent as there are CHILDREN! They're ALL so different! Just keep up the attitude of WANTING to do it 'right'!
P.S. In case you see this. I think that it's usually the 'Melancholy' (negative, perfectionist) personality/temperament type that usually has this 'syndrome'. Like someone else said, he's too hard on himself. He really, really needs positive reinforcement for whatever he does RIGHT and as little criticism and 'negative attention' as possible. Maybe instead of 'punishment', only use 'disciplinary' measurements (milder consequences) with him. Also, he probably is more of a 'team player' than an 'individually competitive' spirit.