Sir Freddie Laker was really the original budget airline maestro, but was undone by events surrounding another carrier - as everyone now knows. (a lot of people were sued in the process, which is why I didn't name the airline).
The problem is really related to how low the price would have to go to attract punters away from current economy carriers. In Sir Freddies day, transatlantic fares had previously been very expensive, but with more 'competition', fares in economy (the cheapest ones) are now really pretty low for a long flight, i.e returns for 199 quid.
I'm not saying they couldn't go lower, just that the difference would not be so great as was seen in Sir Freddies time. On transatlantic routes, all airlines cherish the business/first punters as they are the really profitable ones. This suggests that economy tickets are near as low as they could go (but if anyone knows differently, feel free to correct me)
Following Freddie of course, Virgin gave other transatlantic carriers a real run for their money (if I remember correctly, their starting price at launch for NY return from London was £99), but that is very well documented in the annals of aviation.
The absence or presence of food really does not make much difference to an economy ticket price.
In the case of budget airlines, it is more to do with the way in which the airline is run,i.e selecting the more profitable routes, cutting pay, perks and so on and operating point to point, only having a single aircraft type, no tickets, etc..
The other factor is charter flights, and they probably are the cheapest possible. So goezytransatlantic or whatever, would probably not be able to undercut them by much.
I don't mean to be cynical, but fly economy transatlantic with some carriers nowadays, and you will think you ARE flying budget!