There are a couple of reasons that seem to be at play here.

The biggest is that his association with the song predates the special. But so does his soft spot for kids.

The song "Frosty the Snowman" was written by Walter Rollins and Steve Nelson in 1950, allegedly an attempt to recapture the magic of the earlier novelty Christmas song "Rudolph the Red-Nosed Reindeer." "Frosty" was first recorded by Gene Autry, who had a hit with "Rudolph" the previous year, but later in 1950, beloved vaudeville comedian Jimmy Durante recorded "Frosty" as well. His version reached No. 7 on the U.S. pop singles chart.

So when, in 1969, Jules Bass and Arthur Rankin Jr. were looking for a narrator and singer to appear in their "Frosty the Snowman" animated special, Jimmy's association with the song made him a natural choice. As for why he accepted when, as you point out, he wasn't a big voice actor (indeed, "Frosty" is his only voice role), it seems that has more to do with his sideline as a genuinely good guy.

Durante was a supporter of children's charities, including, most notably, the work done by the Fraternal Order of Eagles to support children with disabilities. In 1961, he was invited to appear at an Eagles convention but refused to be paid for it. When asked what the organizers could do for him instead, Durante said (and you can just hear him saying it), "Help da kids."

In 1967, two years before the "Frosty" special, the Eagles established the Jimmy Durante Children's Foundation in honor of his support over the years.

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