An important point relevant to this question is that the earliest meaning of the word "broom" in English, dating to **c1000**, referred to a particular species of flower.  It appears that "broom" as a botanical term in English was not a metaphorical extension of tools for sweeping, but rather the original meaning, figuratively extended to refer to the tool.

Per the *OED*:

>**A shrub**, *Sarothamnus* (or *Cytisus*) *Scoparius* (family Leguminosæ), bearing large handsome yellow papilionaceous flowers; abundant on sandy banks, pastures, and heaths in Britain, and diffused over Western Europe. Also the genus to which this belongs, and the allied genus *Genista*, including the White Broom, and Giant or Irish Broom cultivated in gardens, and many other species.

The *OED* proceeds to offer an extended definition that includes more plants as definition 2:

>Entering into the name of various other plants used for sweeping, or in other respects fancied to be akin to the broom proper; as *butcher's broom* **n.**, *Spanish broom* **n.**   (a kind of grass).

Note that in this context, "broom proper" likely refers to the plant [*Sarothamnus Cytisus*][1], not "a utensil for sweeping things," which the word would come to mean later, first recorded in the 1400s.

Far more botanical brooms are discussed in modern texts, [including in Wikipedia][2].

Because of this evolution of the word "broom," the *OED* makes only a passing reference to the specious etymology provided by *The Rural Cyclopedia*, edited by John Marius Wilson, which contains this excerpt on "butcher's broom."

>The whole bush is gathered by butchers, and made into besoms for sweeping their blocks and shops; and hence it obtained the name of **butcher's broom.**

The complete text from the journal [can be viewed here on Google Books.][3]

It seems notable that this citation is from **1847** whereas the term "butcher's broom" dates back to **1538** (and, as noted above, **broom** itself dates back to c1000), so the veracity of the *Cycolpedia*'s etymology shouldn't be assumed infallible.

Prior to offering the 1847 citation as comparative evidence, the *OED* makes its more primary etymological assertion:

>Apparently so called on account of the appearance and texture of the stiff, spiny tips of its leaves


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Long story short?
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That long-handled implement that witches and wizards play Quidditch on was likely named after a kind of flower, not the other way around.


  [1]: https://en.wikipedia.org/wiki/Cytisus
  [2]: https://en.wikipedia.org/wiki/Genisteae
  [3]: https://books.google.com/books?id=cOIxAQAAMAAJ&lpg=PA550&dq=The%20whole%20plant%20is%20gathered%20by%20butchers&pg=PA550#v=onepage&q=The%20whole%20plant%20is%20gathered%20by%20butchers&f=false