I'm not trying to defend the warts of the US banking system, but...
"Credit card" would be a third account that is purely debt-based. It's optional, and you can spend directly from your checking account with a "Debit card" instead. It's just not advisable, due to the broken-ass card system you described.
"Savings account" is optional, but earns interest. Does your single EU account earn reasonable interest? Long term deposit ("CD" and "Term Share") rates are currently around 3%, with savings paying 2%.
Combining checking and savings accounts can be done in the US in 2018. The 3 major online banks all allow you to setup an automatic "overdraft transfer", so you can deposit into your savings account, spend out of the checking, and have money automatically move between them. The remaining caveat is that you're limited to six of those movements from the savings per statement, which makes aggregating small transactions on a credit card handy.
I expect the caveat with this Robinhood offering will be a cap on the balance that earns 3%. You can find plenty of checking accounts with high nameplate interest (search for Kasasa), but they all demand some level of ongoing mindshare activity [0] and limit the amount they'll actually pay out, rather than functioning as true savings/moneymarket.
[0] Usually a certain number of debit card transactions per month, to better feed the surveillance databases.