You are assuming that the lettuce-funded buger business is the only possible way to provide free food for those in need. Maybe there are government bugers that are just as good but the burger joint they are served in is not as flashy. Maybe there are people who just like making burgers or want to help others and give them away without insisting on you eating lettuce - or they would if the lettuce-funded burger business didn't have exclusive bun contracts with all the local bakeries that they finance with their lettuce income.

Also, don't forget that big lettuce is not subsidizing the burgers out of the goodness of their hearts - they are doing so because they believe it will allow them to capture more wealth overall. Wealth that people could have used to pay for food.