According to Rav Ya'akov Emden, yes (but that might not mean you have to do it).
He considers "systematic" pronunciation and "dialectal" pronunciation (havarah?) inextricably linked. At the very beginning of the Sulam Gadol introduction to his sidur, the first chavak "explains the quality of the proper pronunciation". Relevant highlights include
צריך להיות שלם בחיתוך השלון, כל מום לא יהיה בו - באות, בניקוד, בטעם.
[One] must be complete in lingual precision, no blemish should be present - in letter, pointing, or accent. . .
Particularly not reading 'ע's as 'א's. . . Unlike we Ashk'nazim do with the weak 'ת' which we shamefully read like a 'ס'. Though we do great with vowels, unlike the S'faradim that don't distinguish between kamatz and patach. . .
He goes on to encourage the prayer to differentiate between the various vowels which have a tendency to be conflated, as well as between the varieties of sh'va, not to join words across phrase breaks, to pause appropriately, and to emphasize the correct syllables. But he concludes with a mitigation, which might explain the disparity between viewpoints in the question's motivating anecdote:
ובמי שאפשר לו ויודע הדברים אמורים
אבל מי שלא הגיע בידי מדה זו בנעורים
א"א להטריחו בהנחת הטעמים שטורחו רב ומייגע דבורו
והפסדו מרובה משכרו
לכן טוב שלא לבלבל הקורא אשר לא נסה באלה
Paraphrasing, one who grew up learning these principles and internalizing them should implement them. But they are too burdensome to be taken on by those who didn't, so they are better off just leaving them alone.
The above quotes, as well as almost the entirety of the massive introduction to this sidur, are in rhyming couplets in the original.