I'd always thought this, but didn't think anyone else out there was sufficiently interested to bother writing about it: very likely, the word "leech" to describe the blood-sucking animal comes from the Old English word for "physician," not the other way around.
When I first began studying Anglo-Saxon magic and medicine, I wondered why the big compendia like Bald's Leechbook and Leechbook III (both in MS Royal 12 D.xvii, for those who care about manuscript stuff) didn't have cures that involved the physicians applying blood-sucking leeches to the patients; it was only later that I realized that the use of the wormy leeches in "leechcraft" only came much later.
Looking at the OED evidence, my best guess is that the two words happened to be very similar in the Old English and, depending on the dialect, were homophones or near homophones. Later, in the very Early Modern period, when physicians started bleeding with leeches (a return to a classical Mediterranean practice), the similarity between the two words combined them.
By the way, I would just like to point out that this is another example of Modernist anti-medieval stupidity. Moderns like to elevate the classical era as a golden period, and the Modern era as the "Renaissance" from the "Dark Ages" -- but bleeding to balance the humours is a classical practice that seems to have been largely suspended in the medieval era, even though medieval practitioners knew about Galen's four humours, and that was only resurrected in the Modern era.
So, in conclusion, the Modern era sucks (blood), and the Medieval era rules.
h/t Moyen Age