How does one pronounce "hath" and "doth"?

Old English verb forms such as "hath" and "doth" are, I believe, normally pronounced with the /θ/ sound as in the word "think."

But somebody once told me that that is actually a mistake. The words, said he, were originally pronounced as present-day "has" and "does" with a /z/ sound. Pronouncing "hath" as /haθ/ rather than /haz/ is a modern mistake, said my source.

Is this true?

Certainly not. I’ve never come across an Old English primer or grammar (I’ve used five or six of them at least) that says anything of the sort!

OED has the following for the third person singular of have:

3rd pers. sing. has /hæz/ , /həz/ , /əz/ , orig. north.; contracted 's (colloq.); arch. hath /hæθ/.
OE hafaþ, hæfeþ, OE–ME hæfþ, hafeþ, (ME afeð), ME hafð, haueð, habbeð, ME haþ, ME hafueð, hæfueð, hæueð, hauið, (aueþ, abbeþ, aþ), ME heþ, ME–16 (17–18 arch.) hath, (ME avyth, hat, 16 haith).

Note that many of those forms are actually spelled with eth and thorn, indicating that the th was the normal pronunciation. It appears to have changed around the end of the seventeenth century — or at least, hath is noted as archaic during the eighteenth and nineteenth centuries.

My English Professor, an Englishman by birth and education, insisted that "hath" ought to be pronounced "haz" and insisted that the prononciation "haθ" was never used. Perhaps he was wrong. Yet I am still not convinced on just how the "-th" suffix ought to be pronounced when it is encounted today because if the people of the 17th, 18th, and 19th centuries pronounced it "haz" would that not make "haθ" incorrect in the 21st century?