I think you mean something like system testing, but backed by BDD-style specs?
TBH, I personally think you're being a bit too hard on unit tests - the number of times they reveal bugs in our our code bases (and in my own code - the horror!) is ridiculous. I do find them really valuable.
That said, testable OO code (I'm primarily a C# guy) has certain constraints, and sometimes making it testable results in a high level of abstraction - so much so that individual tests almost don't seem to actually test anything meaningful, and it becomes difficult to see where the logic and behaviour is, without digging through 42 layers of abstract classes, interfaces and factories.
Recently I've come to favour a kind of inverted test pyramid, where instead of unit tests forming the most substantial foundation, system tests do instead, followed by integration tests, and finally by unit tests at the tip. I find this leads to a "sensible" level of abstraction, where unit tests are used where they are most valuable, and system tests keep tests meaningful. Depending on your code, it might be quite tricky to setup systems tests, and it might need quite a large time investment, but IMO it's worth it. If you're able to dockerise your entire solution, then it's much easier to do.