Is the sliced meat of a gyro in the US not all lamb? A few additives/spices/fillers maybe, but it’s all lamb meat in the UK, and we call it a doner kebeb. Certainly no other meats, unless you go specifically for a chicken kebab. The chunks of un-ground lamb in a pitta with salad is what we call a shish kebab. Like you, I prefer the gyros/doner experience. Around $6 to $9 in London.
Actually, some of the outlets are mildly questionable, so maybe a bit of donkey gets in there.
How about home-made!:
