It is common advice and I've been reading it for decades. But it seems bizarre and unnecessary to me, as if some person who does not appreciate bees, butterflies, or hummingbirds decided to spread this notion. Or maybe concocted out of nothingness by someone who wanted to have "facts" to share about Coleus, or possibly someone who did not understand the difference between perennials being grown in locations where winter is too cold for them to survive perennially vs. true annuals.
Nobody offers this advice for the blooms of Swedish ivy (Plectranthus verticillatus) or Mona Lavender, or so many other nearly identical blooms.
On some truly annual plants, which are those that can't survive perennially anywhere, it can be possible to delay the inevitable death of an individual plant by removing the inflorescence, but that has nothing to do with Coleus because it is not a true annual. It is a tropical perennial shrub.
When I bring Coleus inside for winter, I remove the blooms because they drop the little petals but I always leave them alone on outside plants. There's no inherent reason for us to perpetuate this strange notion about Coleus blooms. It makes more sense to me to more fully educate ourselves past this old notion that doesn't accomplish anything except to make some gardeners feel guilty about not doing it. If one wants to, there's no reason to not embrace these blooms that the pollinators love so much.
If one decides independently that they just don't like the look of the Coleus blooms, please remove them. I don't personally believe it should be done just because someone told you you're supposed to - to conform to their aesthetic preference, or because your plant would be negatively affected in some way if you don't. The latter is not a valid concern and the former may not match your aesthetic preference. Unless you just want to, there's no other reason to do it.
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