This assumption is not correct, and is being propagated as fact, when it is not true. Yes, old school teaching made an assumption that DNA was more robust than RNA, because it is one strand versus two. If that were the only consideration, yes it would be true. But when you take into account the types of replication mechanisms involved and if they were subject to proofreading by the polymerases during the process, the assumption breaks down.
Replication FIDELITY is not just determined by single vs. double strandedness (and RNA in some instances is doublestranded, FYI), but is also determined by whether or not a viral replication has a robust proofreading mechanism built into it and the efficiency of that proofreading mechanism in the polymerase reading the DNA. SARS-CoV-2 has a LOWER MUTATION rate than influenza (which trends to about one major antigenic shift per year) because it actually has a proofreading mechanism built into it's polymerase. Influenza, by contrast, does not.
Relevant articles:
Coronaviruses: an RNA proofreading machine regulates replication fidelity and diversity. - PubMed - NCBI
Mutations can reveal how the coronavirus moves—but they’re easy to overinterpret | Science | AAAS (this is an editorial, not original data, but the references are sound).
End result:
SARS-CoV-2 is seeing slow antigenic drift (i.e. random mutations) at a rate about 1/4 that of Influenza (which doesn't have proofreading built into it's replication mechanism). It's not quickly mutating.
Why is this important? Because it means if we can get a vaccine out there, we have an excellent chance of wiping this virus out of the population.