Why does English have the word "broomstick"?

Oxford Living Dictionaries' dictionary of North American English defines broomstick as :

1 The long handle of a broom.
1.1 A brush with twigs at one end and a long handle, on which, in children's literature, witches are said to fly.

There are other tools with long handles which do not get a dedicated name: shovelstick, rakestick, etc. Moreover, a broom without a stick is a brush, so the differentiation seems unnecessary. Why did the handle of a broom get its own name when so many other tools didn't?

Evidently, the English word broom originally referred to a type of plant that people used to supply the working end of a sweeping device. Merriam-Webster's Eleventh Collegiate Dictionary (2003) treats the definition involving the plant as older than the definition involving the implement:

broom n {M[iddle] E[nglish] fr[om] O[ld] E[nglish] brōm; akin to O[ld] H[igh] G[erman] brāmo bramble} (bef[ore] 12[th] c[entury]) 1 : any of various leguminous shrubs (esp. genera Cytisus and Genisia) with long slender branches, small leaves, and usu. showy yellow flower; esp : SCOTCH BROOM 2 : a bundle of firm stiff twigs or fibers bound together on a long handle esp. for sweeping.

In the San Francisco Bay Area, where I live, scotch broom grows wild in the hills and parklands round about, having been introduced (and having escaped cultivation) many decades ago—and I can tell you that a bunch of its strong, flexible twigs bound to a study wooden stick would make a formidable tool for sweeping.

In the old days, I imagine, broomsticks tended to last longer than broom twigs, and so had a household identity of their own. But we do have comparable terms for the sticks associated with rakes (rake handle or rake pole), shovels (shovel handle or shovel shaft), and axes (ax handle or ax haft), for example, so the implication of the original question that broomstick is unique among hand tools does not survive serious scrutiny. We might as well ask, "Why do we have ax handles, shovel handles, and rake handles but not broom handles?" (Of course, broomsticks are sometimes called broom handles, so that question is problematic, too.)

Undeniably, people sometimes talk about a witch riding on a "broomstick" even though what they have in mind is a witch riding on a complete broom—stick and twigs or fibers. But the Eleventh Collegiate doesn't endorse that meaning of broomstick. Instead it offers this brief definition:

broomstick n (1663) : the long thin handle of a broom

It follows that, as far as Merriam-Webster is concerned, when people refer to a witch riding on a broomstick, they are describing a witch riding on the long thin handle or pole by itself, with no twigs attached. Otherwise, the witch would be riding on a broom, not merely a broomstick.