When programmers really embrace dependency injection like the parent comment, the implementation largely is the inputs, and your code expects them all to be implemented correctly (or mocked correctly by test code). I completely agree that mocks are overused, and their importance in 2008 were in my view a direct result of the popularity of dependency injection patterns.
Following that logic, the best way to remove mocks from your tests is to minimize dependency injection, relegating it all to a single place in the code wherever possible, and implementing all other domain logic as pure functions of input to output. How to test that code which touches other systems? Integration tests (or multi-system tests or whatever you like to call them).
In 2008, I actually found code often got more testatable when it was dependency injected. I spent a month ripping singletons out of a gui application so we could make it into a web service, that would have been a lot easier with a DI model. The system was nigh-on untestable until we got rid of those singletons.
I link it in another comment but https://github.com/gaffo/CSharpHotChocolateZeroQLIntegration... is how I do things these days, api integration test, and more focused unit tests where you don't understand the library well yet, there's lots of tricky edge cases, or error conditions such as setup that are harder to integration test. I'm still feeling it out.
In many cases mocks are now over used where previously they were important in say 2008. Especially now with languages that support functions as objects, better generics, and other features which weren't common a while back. Likewise frameworks are and languages are generally way more testable now which means you're doing less backflips like static injecting wrappers for DateTime.now into libraries to make tests work. This further allows more contract and less implementation specific testing.
As with most things there is a lot of nuance/art to doing it well and smoothly