People who think the It's Coming Home stuff was actually about people believing they were going to win the tournament need to take a look at themselves. I live here and didn't meet a single person who actually thought they'd win the tournament, even after the draw opened up. Even going into the semi-final, people talking about the final were mainly about saying who they would lose to in the final.
The song, the tag line, the memes etc were simply about England having something to get excited about for the first time since Euro 96. And it was very English in it's self-deprecation. It was in many ways the English, and I include myself in that for the sake of this discussion, taking the mickey out of themselves.
I think more than intent, the perception is the problem here. While intended as self deprecating and silly by certainly the vast majority it only takes a few dickheads to ruin it for everyone - and lets be clear that is certainly ratio here, it's a minority that have made it grating.
I personally watched England fans harass and threaten tourists on Courtney Place because they wouldn't answer when asked "is it coming home". The tourists had no clue what was going on and barely spoke English. By the time I was there the police had already been called by bystanders because it was so horrible. I guarantee the lady in tears found nothing about that experience to be "banter" as many seem to claim under similar circumstances.
Many non-English supporters when they hear that song/chant are not in fact reminded of a humorous little ditty but it conjures images of fans trashing Ikeas, jumping on ambulances or some personal run in they have had with a fan being a dickhead about it. And trust me those experiences and perceptions stick in the mind much stronger than the good natured ones.
The same way a few dickheads give league fans or All Blacks fans a very bad rep in some quarters, a few English expats spread around the globe egging each other on have given a certain impression and vibe to that chant and I for one totally understand why a big number of people find it arrogant and smug to listen to - having lost all it's intended meaning.
That incident C Place sounds pretty shark. Reminds me of Viz the English cartoon magazine (popular in my youth) and a Geordie character called Sid the Sexist.
https://en.wikipedia.org/wiki/Sid_the_Sexist
One time time Sid goes to the south of Spain, and overhears some German speaking tourists talking, and starts chanting "1 world cup, 2 world wars".
The tourists tried to explain to Sid they were Swiss German, and didn't understand why he was yelling at them!
But really problems like this are always caused by a stupid minority. Were these English fans in Wellington, young males and drunk? No excuse of course, but if so - always a bad combination. On my 1997 van trip OE around Europe, I saw plenty of fellow Kiwis do some really dumb acts, abusing locals etc that could be disgusting, and totally embarrassing/cringe worthy.
I have also a friend who is rugby writer for one of the major NZ newspapers. He basically hates NZ rugby fans. One time in Brisbane, having a quiet beer, he hid behind a pole, to avoid his own black clad brother (who had travelled over from NZ with a bunch of his mates for a Bledisloe game) spotting him - such is his contempt for the tiresome 'pack male rugby culture'.
So these sort of incidents are hardly a unique English football fan problem.