Thanks @TIFitis for starting this RFC.
I believe in general, end to end testing is not encouraged in llvm-project repository. There are some lengthy threads on this subject in RFC: End-to-end testing. LIT tests are generally supposed to test the specific functionality or pass that a patch adds. I believe this is specifically to keep the tests simple, reduce dependencies (eg: not to cause a failure in a flang test when an LLVM pass changes).
However, there are some projects like MLIR (https://github.com/llvm/llvm-project/tree/main/mlir/test/Integration) and openmp offloading (https://github.com/llvm/llvm-project/tree/main/openmp/libomptarget/test/offloading) that tests end-to-end inside the llvm-project.
Given that some projects already test and the fact that OpenMP testing does depend a lot on other factors (MLIR, FIR, OpenMPIRBuilder etc). A reasonable testing has to ensure that it works with all these components. It would be good to have some end-to-end testing (Fortran to llvm-ir) and exeuction tests somewhere in the Flang test directory.