Ah, I see this is the question where all the cool people hang out! Νικόλαος Στεφάνῳ, Δημήτρᾳ, Μιχαήλ, Ἰωακείμ, Βενιαμείν, Ῥοβέρτῳ τε ἐρωτήσαντι, εὗ πράττειν.
Related question, with rationales: What are the pros and cons of the Erasmian pronunciation?
When I am on my own, I actually mutter Ancient Greek aloud to myself, to try and work out what the hell is going on. (I’ve never actually learned Ancient Greek formally.) And when I’m muttering, it is of course in Modern Greek pronunciation. Which is motivated by familiarity, since I speak Modern Greek.
When I speak aloud to others, it is in reconstructed, because that’s what the word was as far as I know, and reconstruction makes Greek spelling make sense. In fact, I’ve had Greek classicists tell me “could you please stop using Erasmian to me?”
I don’t use Erasmian, because I find Anglicised or Germanicised Greek distasteful—Dzoys for Zeus indeed! But since I never learned Ancient Greek in an English-speaking classroom, I don’t have the pressure of Erasmian familiarity that Anglo classicists do. If I did, it would make much more sense for me to do so.
I occasionally try to do pitch as well. But no, it does not come naturally, and there aren’t many good models for it. (Search the phrase “Yodelling Martians” on Quora for more on what I think of it.)