. "If you cannot survive 2-3 hours of a kids party without alcohol then bring your own or don't come."
A kids party is precisely the occasion when a glass of wine is very welcome! Loads of Screaming kids and paranoid parents, unless there are only a few kids there they're pretty hellish usually.
As far as BYOB goes, it's bring your own bottle. Up to you if you want to take alcohol or soft drinks. If I had an invitation with that on, I would expect that whatever I took would be for me to consume and not to have to give to other guests because the host hadn't provided the right things! At a party that we host we always get in some beer, wine and soft drinks. It wouldn't usually be enough to cater for everybody because in our family everybody usually brings a bottle of wine or something to contribute to the overall stash of stuff so there is plenty to go round. If anyone was drinking something very specific they would usually make it clear when they arrive that what they'd brought was for themselves. I wouldn't think them tight as while they're consuming their own stuff they're not consuming something I've bought so it's swings and roundabouts.
OP, your host should have provided teas/coffees and minimal soft drinks. I think you were generous to share your coke around - the people who wanted some should just have had a cup of tea or water instead rather than nab yours!
Mind you, you can never please all your guests as a host. We provided everything one Christmas party and SIL refused to have any drink she was offered because the mixers we had were not diet ones and there were too many calories in the wine and beer. You'd think she'd have let it go on that one occasion and just had normal coke but she stood there saying "it's fine, I'm ok with water" when clearly it was awkward and DH had to go out and buy some Diet Coke just for her.
If people are on a strict diet they should make sure they take appropriate food and drink to a party because we don't do diet versions of things in our house as I don't believe in diet industry stuff.