The gemara says that the "lamed" implies that one is making a bracha on a mitzva that they are about to do, however, the language of "al" is a machlokes whether it has an implication of a mitzvah that one is about to do, or whether it is more applicable to a mitzvah that has already been done. Another implication that the gemara assumes is that if made with a "lamed" the implication is that the obligation is specifically on the person making the bracha more than anyone else, whereas the language of "al" is more appropriate when the person doing the mitzvah is not the person on who the primary obligation falls. When the gemara concludes that on biur chametz we make "al", the gemara is paskening like the opinion who says that "al" also implies a mitzvah that you are about to do. Rashi understands that the gemara also retracts from the second assumption and holds that even if the one making the bracha is the one who is has the primary obligation, he still makes "al". Rambam disagrees and maintains that when the father does the milah or someone shechts his own korban pesach, he makes la'mul and lishchot, not "al hamilah" and "al ha'shechita".
Based on these rules that a bracha made with a lamed is always done before the act, Tosafos cites the Rashba who says that the father of a child getting a bris should make the bracha of li'hachniso prior to the milah. Since li'hachniso is made with a lamed, it must be made before the bris. Furthermore, the gemara 7b says that all brachos should be made prior to doing the mitzvah. Rabbeinu Tam disagrees because the gemara in Shabbos implies that li'hachniso is made after the milah. Rabbeinu Tam explains that the nature of the bracha of l'hachniso is not a birchas hamtizvah on the mitzvah that is presently being done, rather it is a birchas ha'shvach to praise and thank Hashem, and its function is to show that the bris is being done li'shma, not for medical purposes or chas v'shalom avoda zara purposes.
Presumably, Rabbeinu Tam's approach explains why the bracha doesn't need to be made prior to the mitzvah both from the perspective of the language "li'hachniso" and from the perspective of all brachos need to be made prior to the mitzvah (כל הברכות מברך עליהן עובר לעשייתן). However, Tosafos adds that the reason it doesn't need to be made prior to the mitzvah is that the bracha is only made before when the one doing the mitzvah makes the bracha, but here since the father is not actually doing the bris, there is no halacha of כל הברכות מברך עובר לעשייתן.
The question is why Tosafos needs that addition. Once Tosafos establishes that this is not a birchas hamitzvah, it should follow that it doesn't need to be made prior to the mitzvah even if the one doing the mitzvah makes the bracha. Clearly, Tosafos holds that being a birchas ha'shevach and not a birchas hamitzvah only solves the language issue of using a lamed "li'hachniso", but doesn't address the issue of making brachos prior to the mitzvah because even birchas hashvach should be made prior to the mitzvah. However, the Maharsha points out that the Rosh uses the sevara of being a birchas hashevach to solve the problem of עובר לעשייתן also. Therefore, there is a machlokes Tosafos and Rosh whether birchas ha'shevach need to be prior to the mitzvah.
Tosafos proves that the bracha of li'hachniso can be made after since it is made by someone else from birchas eirusin that we make after since it is made by someone else. The Rambam (Ishus 3:23) holds that birchas eirusin is a standard birchas hamitzvah and is therefore made before the mitzvah is done. The Rambam writes that making the bracha after the kiddushin is a bracha l'vatala. The Ra'avad seems to agree that it is a birchas hamitzvah but says that it should be made afterward since there is a concern that they will not go through with it. Similarly, li'hachniso is made afterward since there is a concern that the mohel won't do the bris. The Ra'avad essentially says that same as Tosafos that when someone else is doing it you don't need to make the bracha before, except that the Ra'avad gives a practical reason for this that we are concerned that they will not go through with it.