Why Only DNA can Contain Hereditary Material?

I think any discussion of this question can benefit from a historical perspective. For a long time, it was in fact believed that proteins was the hereditary material. The Nature Scitable page on the discovery of DNA (1) starts with the following passage:

In the first half of the twentieth century, Gregor Mendel's principles
of genetic inheritance became widely accepted, but the chemical nature
of the hereditary material remained unknown. Scientists did know that
genes were located on chromosomes and that chromosomes consisted of
DNA and proteins. At the time, however, proteins seemed to be a better
choice for the genetic material, because chemical analyses had shown
that proteins are more varied than DNA in their chemical composition,
as well as in their physical properties.

While perhaps easy to dismiss in hindsight, it is possible to understand the reasoning of the day. The "central dogma" of molecular biology, that genetic information flows from DNA to RNA to proteins was only described later, and explained how the production of complex proteins consisting of 20-odd amino acids can be directed by a polymer consisting of only four nucleotides. This allows great complexity and variety in phenotypes while maintaining simplicity of the genetic material. The central dogma also facilitates the seperation of the use (through protein expression) and storage (as DNA) of genetic material.

As Watson and Crick noted in their famous paper, the double helix of DNA, when discovered, immediately suggested how the genetic material could be elegantly copied. While the more complex structure of proteins would likely require a more complex copying mechanism, the specification of amino acids through three-nucleotide codons in DNA allows the regularity of the DNA material to be retained for easy replication while allowing complex proteins to be produced.

However, even though DNA is the primary genetic material today, the situation may have been different at the time life first appeared. According to the RNA world hypothesis, RNA may have been the original genetic material and that DNA is a variation of RNA, not the other way around as it is commonly seen.

For a fuller historical perspective, I recommend the book "What is Life?" by Erwin Schrödinger, which was written before the discovery of DNA as the genetic material.