I have found three (somewhat overlapping) reasons.

1. The custom of the inhabitants of Israel at the time the first Ashkenazi communities were starting was to say tefillot geshem v'tal before the amidah of musaf.

   This is attested to in the writings of R. Shmuel Aboab (1610-1694) where he writes in Dvar Shmuel 149:

   כמו שעושים בקהילות א"י וסביבותיה ... שאומרים הפיוטים ההם קודם תפילת מוסף

   As they do in the communities in Israel and the surroundings ... where they recite the piyyutim [of geshem and tal] before musaf.

   [Note that this is indeed the common custom among eidot hamizrach today, see e.g. what R. Melamed writes here.]

   As was the case in with a number of other practices, the founding Ashkenazi communities in Israel adopted this custom.

2. R. Tzvi Pesach Frank (Har Tzvi, Orach Chayyim 55) notes that reciting these prayers before musaf is consistent with Ashkenazi practice in Jerusalem [and elsewhere in Israel] which in general avoids reciting piyyutim during chazarat hashatz, due to the problem of interrupting the fixed text of the blessings.

   This in line with the teachings of the Gra, whose students were heavily represented among the founders of these communities.

3. With respect to tefillat tal in particular, R. Frank (loc. cit.) connects this practice to the Israeli Ashkenazi practice of reciting morid hatal through the summer (unlike in chutz la'aretz where it is not said at all).

   In Israel, Tefillat tal before musaf functions as the necessary announcement to the congregation to begin reciting morid hatal in their silent amidah. In chutz la'aretz there is no way to announce an omission of morid hageshem, so the congregation still says mashiv haruach umorid hageshem in the silent amidah of musaf, and begins to omits it only from minchah.

I have found three (somewhat overlapping) reasons.

1. The custom of the inhabitants of Israel at the time the first Ashkenazi communities were starting was to say tefillot geshem v'tal before the amidah of musaf.

   This is attested to in the writings of R. Shmuel Aboab (1610-1694) where he writes in Dvar Shmuel 149:

   כמו שעושים בקהילות א"י וסביבותיה ... שאומרים הפיוטים ההם קודם תפילת מוסף

   As they do in the communities in Israel and the surroundings ... where they recite the piyyutim [of geshem and tal] before musaf.

   [Note that this is indeed the common custom among eidot hamizrach today, see e.g. what R. Melamed writes here.]

   As was the case in with a number of other practices, the founding Ashkenazi communities in Israel adopted this custom.

2. R. Tzvi Pesach Frank (Har Tzvi, Orach Chayyim 55) notes that reciting these prayers before musaf is consistent with Ashkenazi practice in Jerusalem [and elsewhere in Israel] which in general avoids reciting piyyutim during chazarat hashatz, due to the problem of interrupting the fixed text of the blessings.

   This in line with the teachings of the Gra, whose students were heavily represented among the founders of these communities.

3. With respect to tefillat tal in particular, R. Frank (loc. cit.) connects this practice to the Israeli Ashkenazi practice of reciting morid hatal through the summer (unlike in chutz la'aretz where it is not said at all).

   In Israel, tefillat tal before musaf functions as the necessary announcement to the congregation to begin reciting morid hatal in their silent amidah. In chutz la'aretz there is no way to announce an omission of morid hageshem, so the congregation still says mashiv haruach umorid hageshem in the silent amidah of musaf, and begins to omits it only from minchah.