I agree. Cheese for a burger is sort of like an accessory, a condiment. In fact, I think it's pretty uncommon that a burger be served without cheese/without the option of cheese. I don't think I've been to a BBQ where burgers were served that there weren't slices of cheese/grated cheese out. If I served chips at my house or roast lamb to my friends on Sunday, I would try to have the "common" or "popular" condiments/accessories out as choices or try and make sure I had them in the house e.g. Tomato Ketchup and Vinegar for the Chips or Mint Sauce for the lamb. If for some reason I had forgotten to offer my guests them, I wouldn't mind in the slightest if they asked for a condiment. If they wanted an unusual one that I didn't have in my fridge I would simply say "no I'm afraid I don't have any of that". When something is so common, its foreseeable people will want it. More people than not would probably like a condiment with their chips so why are they rude for asking? Everyone has ketchup in their fridge. That's what it's there for. Chips and burgers. Most people have cheese in their fridge too and like a sauce, theres no prep for it, it just needs to be sliced or grated. Whats the problem offering it to guests if they ask?
I don't really want to live in a world where my friends are fearful to ask me if I happen to have any ketchup in my fridge that they could put on their burger Seriously? You think its rude to go to your parents house, siblings house, best friends house whatever and ask if they perhaps had any vinegar for your chips? I want my guests to be happy/comfortable and if I have cheese in my fridge and one of my guests asks if I have any I would fetch him some (I'm the host. I don't care if my dish gets cold) or if we were close or I didn't want my meal to get cold I would simply say "ohhh good idea, it completely slipped my mind. Tell you what, there's a block in the fridge. Why don't you get it out and put it on the table so anyone who wants some can help themselves. Thanks for reminding me!" There is a massive difference between asking for a condiment or some inexpensive common topping like a cheese and asking your host to make you a new meal or if you could help yourself to a 'meal' she hasn't offered you. When an ingredient is more common than not to be eaten with a certain dish or at least quite common, I don't think it rude at all someone ask. Even if its not common, they're only asking! A sauce or condiment or common topping is 1. Foreseeable that something people will want and 2. Hardly an imposition or ungrateful or rude behaviour to ask for. It's for a FRIEND or a GUEST. Don't you want to have a close enough relationship with them that they feel they can ask you if you have any ketchup/an extra towel/a safety pin etc...? Asking for an accessory is not the same as asking for a meal or side dish.
If I knew someone (especially a dear friend or family member) had cheese in their fridge and I asked if I could perhaps trouble them for a slice to put on my burger and they said the burgers were not going to be served with cheese so no I could not have some, I'd think they were being really mean deliberately.