(2-part post):
I think that you've summed up your problem by saying " If I could only be sure we wouldn't be turned down here (for the visa) and in the US (by port authority)". Unfortunately, you can't be sure either way.
It has to come down to a personal choice: Do you want to board the plane knowing that there's a chance (however small) that you could be refused entry to the USA on arrival or would you rather get the matter sorted out ('for better or for worse') prior to travel?
Personally, I'd go for the 'official' route and apply for visas. At least you'd know, one way or the other, whether the US authorities will let you enter their country. Of course you risk that you might have your application turned down, in which case you'd lose money and upset your children. The alternative, however, is to risk being denied entry on arrival. You'd be forced to leave (at your own expense) on the first scheduled flight back to the UK (which could cost you thousands of pounds) or being held, effectively in prison, until the date of your return flight (which would be far more distressing for your children).