If automated tests/linters are a part of your build system, then they basically are breaking at compile time. (Having to write tests to mimic functionality of a compiler seems wasteful to me though)
Type checking is a form of proof. It proves your system is type safe. You would need billions or trillions of tests to achieve type safety equivalent to a type checker.
A successful test only proves something for a single test case. In fact it is impossible to do the equivalent of type checking in the runtime code of your program unless the runtime code has the ability to self "reflect" on it's own source code.
There are also complex type systems that can prove correctness and completely eliminate the need for tests all together but this style of programming is really challenging and time consuming.
