Is your contention that every culture is just American culture (perhaps imitation)? There is no other cultures to experience?
Pretty much, yes. If there is native culture remaining under the veneer of American pop culture, you as a typical tourist won't get to experience it, unless it's a staged performance put on for the benefit of the tourist trade. Just turn it around: if you happen to encounter say a Chinese tourist wandering around your home town, are you likely to invite that person home with you so s/he can experience real American culture?
Wow.
I'm not saying you can go discover some hidden culture that's untainted from 100 years ago or something, but to claim that all cultures are just the American culture or a close imitation is, as I said above, laughable.
As far as inviting a tourist into my home to experience real American culture, a few problems with that:
1) I live in Vegas. I think they came to see the glitz of Vegas, not the "real" culture of someone living in Vegas.
2) I think they can experience the real culture of America merely by traveling through it for an extended length of time (the various parts, as I think the culture of Portland is very different than Georgia, New England, or Kansas). Inviting them into a home isn't necessary. Nor is it necessary for you to be invited into someone's home while traveling to experience
their culture; you can do so by traveling in their country.
3) No, generally Americans don't do this (invite a tourist into their house). It does happens sometimes though. But that doesn't mean foreigners won't. Maybe we are just the inhospitable assholes. And they actually might, because ...
it's part of their culture.
4) If you're so hung up on being invited to a local's home as necessary to experience the "real" culture of a place, it does happen. I read travel blogs all the time where it happens. This blog, Drive Nacho Drive, shares an experience of asking for a safe place to park their camper, being directed to a nearby temple and meeting the people:
http://www.drivenachodrive.com/2013/12/santanam/Seems pretty authentic.
Or how about this one:
http://www.drivenachodrive.com/2014/01/the-onam-food-babies/People
literally invite them to their homes. "You should come to our house and stay. You can have Onam with our family tomorrow."
They ended up declining because they were just passing through, changed their minds and regretted declining the invitation, stayed the next day for the Onam ceremony, and then ...
got invited to someone's house (a completely different set of people from the first invitation!). "You should come to our house for Onam!"
I personally don't think being invited into a local's home is a necessary part of traveling, but you seem to think it's important, so there you go. These are multiple examples, all from within the past month or so. There's more I could dig up, on that blog, and others, but the point is: it happens. And it's not infrequent.
American culture might have had an influence on theirs, absolutely. But if you can't get past the fact that you may recognize a song in a restaurant and equate that to "the culture is the same," I don't really know what to say to you. Culture is not just going to see native dances from 1,000 years ago, with people in the same outfits they wore 1,000 years ago. It's the living breathing differences today. And I guarantee you, it's not the same as American culture (which itself isn't all the same).
There is a vast difference between Germany and Greece (let alone even more different places, like the differences between Thailand and Canada), despite the fact that some people in both places may be wearing blue jeans.