Common misunderstanding of Birkhoff Theorem

In GR you use coordinate charts which cover some specified open region of spacetime. You are allowed to choose the region that you want to cover, and it may be all of spacetime or it may be some smaller region.

For a spherically symmetric thin shell, if the chosen region is strictly inside the shell then it is the Minkowski metric. If the region is strictly outside the shell then it is the Schwarzschild metric. If the region contains the shell then it is more complicated, and this complicated region is the focus of the paper.

There is gravitational time dilation between an observer on the interior of the shell and one exterior to the shell, but this is only relevant if you are covering both the interior and the exterior. Strictly within the interior it is not relevant and the straight Minkowski metric is valid strictly in the interior.