Yes, this is almost possible, but only with one bank: Wise.
EDIT: Here below I explain how I understand Wise works because people around me who didn't know about it have suggested similar ideas for international transfers. I believe you allude to the same scheme of 1 bank in multiple countries, and sfxedit's answer also suggests a similar scheme, but with acquaintances rather than a neutral 3rd party.
So the reason to explain it is because yes, it can be that simple.
And yes, this reads like an ad because I am enthusiastic to share about it. It has saved me many times from international transfer fees, and I feel the least I can do is to raise awareness.
Furthermore, I would really like Wise and other neo-banks1 to succeed because I feel the traditional banking has deliberately not pushed new technology to clients in order to keep them paying ; meanwhile, while they have optimised their IT, reduced personnel, and automatised most of the operations.
1 See this article on Forbes about what is a neo-bank. It doesn't have to have branches. Yes, Revolut, N24, Neon and others are neo-banks.
Wise, formerly TransferWise, has set up exactly this back in ~2014, in order to lower fees of sending money across countries. They had accounts with big banks in multiple countries, but this was transparent to the end users. In turn, end users use the Wise app to tell the bank about the transfer: amount, destination, currency, etc. Then they are given an account in their local country (e.g. US in your case), and a reference that tracks the transfer. Users send the money to this account, mentioning the reference. Upon receival, Wise retains a small fee (much smaller than regular banks), and sends money from their foreign account to the user's destination (e.g. Europe in your case).
Now they also offer an "international" card. You can charge it with money in your currency of choice, and then pay and withdraw money in any currency in any country. You can even pay online, avoiding costly conversion fees (there still are some, but are way smaller and the exchange rate is fair).
I am not affiliated with Wise, but I've been a happy cross-country user of them since 2015 and I think they provide a really good service.
P.S. Wise also has a referral scheme: if you join with a referral link, your first transfer is completely free, and the person referring you may get ~€40. "May get", because they only get it when accumulating 3 invites