In the first place, you are constructing a false parallel. Somewhere does not mean “at some [=indefinite number of] places [plural]”; it means “at some [=indefinite] place [singular]”. I read that somewhere, for instance, does not mean “I read that in a small number of sources” but “I read that in at least one place which I cannot (or will not) at this moment identify”.
In the second place, history demonstrates that manywhere is ‘an elegant and logical solution’ to a problem which does not exist. Take a look at the Google Ngram display (I have had Google graph the incidence of manywhere as a proportion of the incidence of manywhere to anywhere, to give a clearer indication of frequency):
Over the past 100 years there's one spike at around 5,000:1, and a couple more at 17,000:1 or less. The 1902-1903 spike is particularly interesting. Henry James floated the word in an 1894 short story (other early hits are false positives), but it was Lord Avebury (Sir John Lubbock), a distinguished banker and politician and writer on archaeology and evolution, who successfully launched it in The Scenery of England, 1902, where he tossed off the sentence ‘Smoothed and polished rocks occur also “manywhere”, if I may coin the word, in our northern districts…’
The coinage excited considerable comment in newspapers and journals; this is representative:
The new word “manywhere” will undoubtedly find as ready application in adwriting as it has in newspaper offices and general literature, for it is a genuinely useful word, giving expression to an idea that has heretofore had no symbol. It was coined by Lord Avebury, the noted British scientist, and used in a recent book on geology as a fillgap between “somewhere” and “everywhere.” While freely criticized by philologers, it is generally thought that “manywhere” will eventually attain a place in the dictionary. —Printers Ink, a journal for advertisers, June 3, 1903
The word had a brief vogue (mainly among geologists, ethnologists and, as Printers Ink predicted, adwriters), but eventually disappeared again. In this respect it contrasts significantly with two other words coined by Lord Avebury: neolithic and paleolithic, which are still in use.
I'll leave it to others to explore the tiny post-WWI and post-WWII spikes; but it’s pretty clear that however useful manywhere may appear in theory, the speaking, hearing, reading and writing public has no practical need for it.
This should not discourage you from employing manywhere as a peculiarly expressive member of your own lexicon. Its meaning is transparent, so you need not fear misunderstanding; and it strikes me as a cheerful sort of word, which could lend a spritely air to your discourse. You might even create a new vogue for it—and perhaps Lord Avebury was a century ahead of his public, and now is the time for the word to enter common parlance.
