Short answer:

There's an entire historical subspecialty devoted to placenames -- its called "toponymy", and it's particularly important for things like tracing Viking presence in England. . . in this case the shift from "aa" to "a" isn't particularly significant, just fit English readers' habits better. The discipline where you first find toponymy is usually called "philology"-- but many historians make use of these techniques.

Discussion:

The Dutch settlement was originally "Nieuw Haarlem", but the "Nieuw" disappeared not too long after the English take control (1664); the English attempted to incorporate the settlement into Manhattan, and to give it the more English name of "Lancaster" . . . the incorporation succeeded, but Lancaster didn't stick. At that point, the "Nieuw" was redundant -- anglophones didn't need to make the distinction between old and new as might a Dutchman-- and so you've just got "Harlem", spelled in various ways.

The double "a" in "Haarlem" is unusual to an English reader-- we don't use double a's often. There is a long tradition of transcribing vowels that don't fit English to something more familiar, for example from the Germanic umlaut with an "e" to render it in English. EG Wilhelm Böring becomes "Bill Boeing"; Göring becomes "Goering" and so on, but the double A poses the problem-- "how are you supposed to pronounce that?".

If you look at early maps, it seems that 18th century English speakers were aware that the a in Harlem was different, but spellings are inconsistent. A 1776 map has it as "Harlem"; a 1777 map has it as "Hærlem". An 1839 map still has it as "Haarlem", with two As.

Place names tend to preserve older language traditions-- people copy over maps without translating. Throughout the Hudson Valley, for example, you'll find place names ending in -kill; that's Dutch for stream or brook. Maps are very "conservative" -- printers used the same plates over and over again for years, making changes, but rarely altering existing spellings on an existing plate.

But odd spellings are a problem that encourages drift to something more expected. An English reader or writer rarely encounters "Aa" unless they're writing a foreign word or rating corporate bonds-- and is unclear how its meant to be pronounced. Is that meant to be a "long a", as in "cape" or a dipthong like "hare", or a short a like "pat"? We write some long vowel sounds that way ( the double e in "sheet"), but not the "a"; it is one of English orthography's idiosyncrasies that we can readily accept "pete" or "peet" or "peat", but a reader couldn't be sure how "faat" should be pronounced: is that "fat"? Or "fate"?

So it looks as though anglophones pretty rapidly decided that they weren't keen on rendering the double aa in Haarlem. Some did, some didn't, wouldn't have made any difference in pronunciation, and multiple spellings persisted through the 18th and into the 19th centuries, but by the end of the 19th, it'd have been unusual to see "Haarlem", except as an antiquarian affectation.

Similarly with "the Bowery" -- that's something that an anglophone can read, spell and pronounce predictably . . . the Dutch was "bouwerij" -- not a spelling that makes sense to someone used to English language spelling conventions.

A look at maps over the years shows a lot of idiosyncrasies in the retention, transliteration or translation of Dutch names. So, for example, there's a stretch of water between the Harlem River and the Hudson that was known to the Dutch, and is known today as "Spuyten Duyvil" . . . but on a 1781 map, it's been rendered as "Spiking Devil" -- that's a mistranslation, "Spuyten" means "spouting" or "spitting", but it appears that this mapmaker was writing what he heard, or copied from what someone might have heard. Washington Irving's 1809 tale A Knickerbocker's History of New York includes a "just so story" where Iriving comes up with an idea that makes sense in English (but not in Dutch) that involves to "spite the Devil".

Sources:

Zelinsky, Wilbur. “Some Problems in the Distribution of Generic Terms in the Place--Names of the Northeastern United States.” Annals of the Association of American Geographers, vol. 45, no. 4, 1955, pp. 319–349.

Hale, Edward E. “Dialectal Evidence in the Place-Names of Eastern New York.” American Speech, vol. 5, no. 2, 1929, pp. 154–167.

Wm. H. Carpenter. “Dutch Contributions to the Vocabulary of English in America: Dutch Remainders in New York State.” Modern Philology, vol. 6, no. 1, 1908, pp. 53–68.