First of all chickens lay eggs, cows provide milk.
Second, we (in the UK) don't eat ponies, dogs, cats, snakes, budgies, foxes, swans, badgers, goats, otters, spiders, frogs, etc etc. Last time I checked none of those were extinct so clearly they have somehow managed to find a way of surviving without us helpful humans.
Third, as a vegetarian, if the only reason for keeping a species in existence (as your somewhat loaded question implies) then, yes, I personally really would rather they were extinct. You are quite correct in one respect, they would not have to suffer anymore. Of course I also don't believe lunch is a good enough reason in itself for allowing other creatures to survive.