Sieve theory is not saturated. It is alive and thriving. The ICM just awarded the Fields medal to James Maynard in no small part due to his work in sieve theory (see here and here). Because of the natural role of multiplicative structure in sieve theory, there are lots of fascinating and important results in the theory of $L$-functions and modular forms that rely decisively on sieve theory. In these instances, it is usually the flexibility of sieve methods that ends up being the key to success. On the other hand, there are lots of things that one can study about $L$-functions and modular forms that have nothing to do with sieve theory.

For a first look at sieve theory, I think that "An Introduction to Sieve Methods and Their Applications" by Cojocaru and Murty is very nice. As Stanley Xiao mentioned above, a more advanced book (but still nice to read) is "Opera de Cribro" by Friedlander and Iwaniec.

Here are some examples where sieve theory, $L$-functions, modular forms, and other ideas combine, resulting in truly splendid results. This list is far from exhaustive.

- Matomäki and Radziwill's work on sign changes for Hecke eigenvalues of holomorphic cusp forms;
- Radziwill and Soundararajan's work on moments of $L$-functions associated to quadratic twists of elliptic curves (where they, in essence, prove a robust "Brun-Hooley type" sieve for $L$-functions);
- Holowinsky's work on shifted convolution sums, which played a key role in the resolution of the holomorphic variant of the Rudnick-Sarnak quantum unique ergodicity conjecture.