Yes.The host was obliged to open his gates to literally anyone who knocked the door.The thing goes like this:A beggar knocks on a door,the host opens and he is obliged to serve the beggar food.After they dine,the host asks for the stranger's name.It was considered rude to ask him his name before serving him food.Greeks didn't only saw this as a good deed for a fellow Greek,but as a good deed to Zeus himself.

You will find it interesting if I tell you that this kind of mentality is still going on in Greece,but mostly on islands and villages.

Would a traveler just go to some random stranger's house and request xenia?

Nope. A rich/famous guy would just go to a richer/more famous guy's house and request xenia. In order to impress the distinguished guest, the host would offer everything to them.

It's not like a regular poor fellow will open his door to a complete freaking stranger who might be a thief for all he knows. People can be very religious, thinking the stranger might be a God and all, but not to the point where they become that gullible.

Regular, widespread Xenia was not a thing in the ancient world. After all, most (if not all) of the sources that mention it are Homer's epics (and some other mythical tales). It's just that we Greeks like to think that our ancient ancestors were the best people that ever existed, to the point where their attributes become ridiculously illogical.