It seems to me that you are asking two different questions.
For the first, it is far from clear that davening before neitz is any problem compared to davening later - which I think you're assuming. The Shulchan Aruch uses a language of "b'diavad", but Igros Moshe (OH 4:6) and R' Isser Zalman (Even Ha'azel on the Rambam on Krias Shema, if I remember right) suggest that that means, compared to davening exactly at neitz or immediately just after. In their opinion, davening before neitz is no different from davening after neitz. I think the language of the Tur (89) is especially clear:
זמן תפלת השחר מתחיל משעלה עמוד השחר והאיר פני המזרח שהיא כנגד תמיד השחר ומאותה שעה ואילך הוא זמנה ומ"מ עיקר מצותה עם הנץ החמה אלא שאם התפלל משעלה עמוד השחר יצא ונמשך עד סוף ד' שעות
The time of Shacharis begins with dawn and the brightening east, as [the prayer] corresponds to the Tamid of the morning. From then onward is its time, but the main mitzvah is [to pray] as the sun rises. But if he prayed from dawn he has fulfilled his obligation, and that continues until the end of the fourth hour.
I see that the Mishnah Berurah (89:1) and others do agree with your assumption.
Later in that section, the Shulchan Aruch (89:8) talks about davening before משיכיר, before one can distinguish. That is a still earlier time and not advised because it interferes with other parts of davening or putting on tefillin.
For the second question, why not move it a few minutes to neitz? Some minyanim indeed do that. But when they asked R' Feivel Cohen shlit"a (the בדי השולחן), the Rav of the shul where I used to daven, he brushed off the idea: Some people daven neitz (the special people called vatikin, which Rashi describes as [Berachos 9b] "humble men who love mitzvos"), and they do it all year-round. The rest don't.
I don't know if other rabbanim agree with that idea or have some other reason. It certainly seems to be the minhag. My current synagogue has hundreds of members and only one Shacharis minyan: 6:30 every morning.