The Mishnah in Chullin 141a says
הנוטל אם על הבנים ר' יהודה אומר לוקה ואינו משלח וחכמים אומרים לוקה משלח ואינו לוקה
Someone who takes the mother bird with the children, R' Yehudah says they're lashed and don't need to send away the mother bird. The Sages say send the bird away and they're not lashed.
The discussion there makes it clear the dispute is what's the status of this mitzvah: לאו שקדמו עשה, a negative mitzvah which was preceded by a positive mitzvah (so there's lashes, as per Makkos 14b), or a לאו הניתק לעשה, a negative mitzvah which was given over for positive rectification (so there's no lashes, see Makkos 14b-16b).
Now, I'm basing my understanding of these two types of mitzvos on how Rashi in Makkos loc cit. explains it (I am aware there are other explanations, I'm just not familiar with them). If I understand him correctly, לאו שקדמו עשה is that the positive mitzvah is possible before the negative mitzvah (ex: a rapist has to marry his victim, and can never divorce her). Whereas לאו הניתק לעשה is the positive mitzvah is only possible after breaking the negative mitzvah (returning stolen property for example). Therefore, the explanation of the dispute in sending away the mother bird based on the that gemarra in Chullin is that R' Yehudah understands "שלח תשלח" is מעיקרא משמע, at the onset, so it's possible before the negative mitzvah. The Sages seem to hold the mitzvah to send her away is only possible to fulfill after transgressing the negative mitzvah.
Therefore it would seem to me that there's no mitzvah at the onset to send away the mother bird according to the Sages, unless the person transgressed the mitzvah of taking them all. Only R' Yehudah says there's a mitzvah to first send her away. Now, I would guess practically it makes no difference, if you want the eggs and don't want to even momentarily break the Torah, send away the mother. But it seems there's no mitzvah lechatchila (at the onset) according to the Sages.
However, it sounds like from various sources that there is a mitzvah lechatchila. I found in Rambam's Sefer HaMitzvos Shoresh 6 that he writes explicitly that sending away the mother bird is a לאו הניתק לעשה, yet it is a positive mitzvah since Chazal call it a mitzvah (I'm guessing an example is Kiddushin 39b). He seems to learn לאו הניתק לעשה differently than Rashi. Does he mean it's a mitzvah lechatchila, at the onset? He lists the mitzvah in Mitzvas Aseh 148 as to send away the mother bird; no prerequisites. How does Rashi understand things? Would he say the gemarra in Kiddushin that says it's a mitzvah is like R' Yehudah? Or am I misunderstanding the principles at work.
I'd like an explanation which summarizes the various opinions. What's motivating this question is many people seem to think there's a mitzvah to go out and send away mother birds, but according to the Sages (who are the final halacha), there's no such thing. It simply rectifies the transgression.
Edit: it should be noted I saw the Tiferes Yisroel addresses this issue and writes (Makkos 3:4 in Yachin)
וחכמים אומרים משלח ואינו לוקה דס"ל שלח גם אחר לקיחה משמע, והו"ל ניתק לעשה:
The Sages say send away the mother bird and you won't get lashes, since they hold the verse implies also to send her away after you take, therefore it's nitak le'aseh.
He seems to hold the Sages agree to Rabbi Yehudah it's a לאו שקדמו עשה, but the way I explained things I feel like that makes it mutually exclusive to being a לאו הניתק לעשה, so I don't understand.