Wikipedia said:In 1972, Walter Rollins admitted in an interview with Life magazine that initially the concept of Frosty the Snowman was written as a cautionary tale pertaining to the scare of "nuclear winter" and initially it was said that nuclear fallout mixed with the snow and children's dreams of a world without war were what brought Frosty to life, but the publishers and Gene Autry thought that the song would have more commercial value as a children's Christmas song.
I guess the broom and scarf are independent of Frosty, meaning someone else can use them? And so Frosty would have to go find the originals if he's killed, his stuff is stolen, and then he's rejuvenated. Is that the idea?
I think the broom and scarf should be emergent properties of Frosty. Any broom he touches becomes a wand of fireballs, any scarf a rope of entanglement. He can only have one at a time though, of course.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.