This was an important experiment, but did not convince everyone.
Not even Hershey and Chase claimed this conclusion, although it convinced many. In most mol bio classes or when reading "The Double Helix" you can see how there were lots of facts pointing to DNA, but doubts lingered. Some viruses contained RNA only and not DNA, but still replicated. There was also a school of thought which felt that proteins could be the genetic material.
It was Watson and Crick that finally convinced everyone that DNA was the medium of genetic inheritance. The reasons for this are varied, one of them is probably that atomic structures are so clearly understandable and seem so uncomplicated compared to a wet lab experiment which is built upon so many other experiments. Questions keep coming up: "how do you know you shook the flask long enough?", "What if a small amount of residual viral protein got in?" For every revelation like Hershey and Chase, there are many cases where someone stopped an incubation too quickly or wrote down an illegible number and was found out after publishing.
And you can see how much the structure gave us in one fell swoop: the structure of DNA inspired a mechanism by which replication of the DNA could be produced with some error correction, the central dogma (DNA -> RNA -> protein). There were several years more effort to demonstrate that the genetic code existed and then parsing out the exact codons and the fact that they were 3 bases long. But all of this was predicted when the structure was found. So much clearer as to what is going on.