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 sugya there makes it clear the machlokes 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 the sugyos 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 there is a positive mitzvah possible before the negative mitzvah (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). The machlokes therefore by sending away the mother bird seems to work with this explanation of the two sugyas. The gemarra in Chullin says 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 is 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 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 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.
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.