In general, fast tests are a must-have, but I find that means figuring out how to write fast integration tests as well so that they can also be run as part of a TDD-like cycle. In my experience, integration tests can generally be written to be very quick, but maybe my definition of an integration test is different from yours?

For me, heavy tests implies end-to-end tests, because at that point you're interacting with the whole system including potentially a browser, and that's just going to be slow whichever way you look at it. But just accessing a database, or parsing and sending http requests doesn't have to be particularly slow, at least not compared to the speed at which I develop. I'd expect to be able to run hundreds of those sorts of tests in less than a second, which is fast enough for me.

I inherited a django project which has mostly 'unit' tests that flex the ORM and the db, so they are really integration tests and are painfully slow. There is some important logic that happens in the ORM layer and that needs to be tested. At some point I want to find the time to mock the database so that they can be faster, but in some cases I worry about missing important interactions. Domain is highly specialized so not very easy to just know how to untangle the mess.