Yeah but the catch is that in order for your code to be "testable" you have to rewrite it completely in a way that comes with large sacrifices, making it much more confusing, abstract, larger and much harder to work with, in every way except testing it.
I totally agree that if you have to choose (and usually you do, unless you have unlimited time and funding), choose to implement end to end integration tests first.
It also helps form thinking about test cases and logic before writing code. Before I used exhaustive unit tests I would just start writing code right away and often times this would result in constant refactoring as the code evolved or during debugging. But unit tests encouraged me to scaffold out abstract functions and write tests before implementing the code. It got me thinking about what the logic and workflows then implementing the functions meant having the tests pass as verification.
Not only does it make your code more unit testable, it makes your code much more ad hoc testable. Again, I would not refactor a large, non-disposable application just to achieve this.