All Ashkenazim (including Israelis) should at least learn and daven (if not speak) with havara Ashkenazis (when speaking Hebrew or Aramaic).
All Jews (including the Ashkenazic Yeshivish) should at least learn and daven (if not speak) with the emphasis on the correct syllables (when speaking any language especially Hebrew and Aramaic).
And I don’t think schools should be teaching talmidim to speak Hebrew with “oy” instead of “oh” because (to the best of my knowledge) that’s not correct according to any havarah.