I had thought that this story in the December 1930 issue of Modern Mechanics and Inventions

…was an indication that Lovecraft’s passing aside about a beetle race in “Shadow out of Time”…

“After man there would be the mighty beetle civilisation, the bodies of whose members the cream of the Great Race would seize when the monstrous doom overtook the elder world.” — H.P. Lovecraft, “Shadow out of Time” (written Nov 1934 – Feb 1935).

… might have been his picking up on a current of thought popular at the time. But there is a much earlier precedent in Lovecraft’s works…

“You and I have drifted to the worlds that reel about the red Arcturus, and dwelt in the bodies of the insect-philosophers that crawl proudly over the fourth moon of Jupiter. How little does the earth self know life and its extent! How little, indeed, ought it to know for its own tranquility!” — H.P. Lovecraft, “Beyond the Wall of Sleep” (published October 1919).

Possibly Lovecraft got the ‘successors to humanity’ idea from H.G. Wells. There is an excised section of The Time Machine which imagines a far future Earth in which the degenerated last men are preyed upon by gigantic insects — although Lovecraft didn’t read that book until his New York soujourn, and since he read the book version he wouldn’t have seen that extracted section with the insects.

Possibly the idea of having the beetle race be “the successors of humanity” was a later addition to the 1919 idea in Lovecraft’s mind. One wonders if seeing people encased in the black carapaces of the early cars didn’t spark his imagination on that point?

Here is Lovecraft on the topic in a letter, in the mid 1920s…

“if the sun gives heat long enough, there will certainly come a time when the mammal will have to go down to subordination as the reptilia went before him. We are not nearly so well equipped for combating a varied environment as are the articulata; and some climatic revulsion will almost certainly wipe us out some day as the dinosaurs were wiped out—leaving the field free for the rise and dominance of some hardy and persistent insect species—which will in time, no doubt, develop a high specialisation of certain functions of instinct and perception, thus creating a kind of civilisation, albeit one of wholly different perceptions, (when other species view a given object, their ocular image of it differs—sometimes widely—from ours) emphases, feelings, and goals.”