
What’s the truth? Genetics predates genomics by a century or more, and DNA by decades. The basics of the field were elucidated by Gregor Mendel in the 1860s. He originated the “laws of inheritance”, though unfortunately his work was ignored by contemporaries. By “laws of inheritance” I mean that Mendel formulated an analytic model that allowed for discrete inheritance and predictions of the outcome of that inheritance. Naive human understanding of heritability usually relies on an intuitive “blended theory”. It works, after a fashion, but it does not explain many patterns we see around us (e.g., recessive expression).

The utility of Mendelian genetics is why the field exploded in the first two decades of the 20th century. Read A. H. Sturdevant’s 1913 paper on the first genetic map. I think it gives you a flavor of the rate of advancement. Genetics was definitely a field. In The Origins of Theoretical Population Genetics Will Provine outlines how this particular field of genetics developed between 1920 and 1940 to become the core of evolutionary biology. Again, this suggests that even before DNA genetics was an important field.

No one needs to be told about how genomics revolutionized genetics. But it’s major impact has been transforming an often theoretical field into a massively empirical one. Modern genomics is still underpinned by the logic of Mendelian genetics. Analysis.
* The main exception here I’m going to make is for agricultural genetics, but much of this work doesn’t need “genes” as such.

The great thing about Mendel’s discrete genetics is that it is easy to imagine a mechanism. Intuitive blending inheritance seems like it should reduce variance, but you could imagine instead using an empirically validated gaussian model saying what happens to mean and variance. You might be confused about what produces this behavior, but if you want to base evolution on it, that’s ok. In particular, Galton’s studies of height in the 1880s could have produced such a model, although I’m not sure he did.
I’ll suggest another exception: description of genetic syndromes. A lot of them were described in that time, once the idea of particulate inheritance took root and the math to test dominant/recessive inheritance became available.