I've got an argument to settle here. Are the longbows of today mostly center shot or not?

There are probably more cut to center than there used to be, but still many of them cut out from center, too.  Most all Hill style and moderate D/R are cut 1/8 inch or so out, while the newer "hybrid" types with more exaggerated D/R are generally cut to center.  You're both right.

Yeah, what Fletcher said.  My Great Northerns, Robertson's and Hills, for example are cut 1/8 from center.  My A&H ACS, on the other hand is cut a good 1/8 inch past center.