You are not correct here, unless you're saying it specifies the prime RBD for any fare. In your example making an Economy class search withwill put LHR-JFK in Business (I) as well. Why? Because ITA is looking for these options:
1) DUB-LHR in I, LHR-JFK in anything - there are no DUB-LHR I fares that allow a combination with LHR-JFK and the DUB-JFK fares require a diff booking code for that leg, and
2) a DUB-JFK fare that allows booking I on any single leg of the journey - obviously the DUB-JFK I fare satisfies this.
(in this case it's pretty straight forward which legs can be combined with what but as the number of flights grow so does complexity, and ITA is damned smart about picking fares for the flights in the itin)

But it doesn't have to be an I fare, it can be any fare that both includes DUB-LHR and allows a single leg to be booked into I. It can probably be better illustrated with United A fares that usually book into C or D ex-US, you specify A on an all economy flight, it will book the economy i Y, the longhaul into A and use a C/D fare.
(and this is where my dual booking class/fare specifying question is relevant, you can't pick First as class of service because some legs don't have it, but need to specify both the booking class for the longhaul and the fare used which is not the lowest. Also with 2+ stopovers and 7 legs on a single fare that uses higher booking codes than the fare code gets pretty complicated pretty fast and even ITA will lose track of what's what and starts breaking the journey.)

It does, kind of, see above. It takes the segments you feed it and starts iterating through valid fares to see which one is the cheapest that satisfies your criteria. In that way it does understand.

You are absolutely right, it works! I don't know what I did wrong when I tried it the first time but it most definitely works. Thank you!