parshanut torah comment - Why does Rashi cite the Midrash of Pharaoh's daughter stretching out her arm?

There's a famous Midrash cited by Rashi (Shemoth 2:5), that says Pharaoh's daughter stretched out her arm and that it extended beyond its natural length to pull in the basket containing the baby from the Nile river (whom she would name Moshe). This is based on the words of the Pasuk, "וַתִּשְׁלַח אֶת-אֲמָתָהּ", which are usually translated as, "And she sent her arm," suggesting that the sending is somehow awkward for a limb. But that's not really the meaning of the words, and Rashi himself first says that אֲמָתָהּ actually means "her handmaid". (See also, "וְיוֹם הַשְּׁבִיעִי שַׁבָּת לַה אֱ-לֹקֶיךָ: לֹא-תַעֲשֶׂה כָל-מְלָאכָה אַתָּה וּבִנְךָ וּבִתֶּךָ, עַבְדְּךָ וַאֲמָתְךָ וּבְהֶמְתֶּךָ, וְגֵרְךָ, אֲשֶׁר בִּשְׁעָרֶיךָ.‏ " -- "But the seventh day is a sabbath to HaShem your G-d: do not do any creative labor, you, your son and your daughter, your (male) slave and your handmaid and your animal, and the stranger that is within your gates.") Even after citing the Midrash, Rashi further explains that the Midrash doesn't even work grammatically, because the word אֲמָה needs a Dagesh in the Mem to mean "arm".

So the Midrash is using the visual of Pharaoh's daughter miraculously extending her arm beyond its natural length to explain why the word "(and she) sent" is used in the Pasuk - even though, if we just translate it the more natural way (ie., her handmaid), it would make sense without any obvious miracles or any unusual linguistic tricks. Not only that, but the forced translation also creates a new awkward phrasing (at least in my opinion). The very next word says, "וַתִּקָּחֶהָ" - "and she took it". Once the Midrash has defined the meaning of the Pasuk that she herself extended her hand, it makes this word awkward, because it could have just said, "to take it," or something equally generic and passive. It's not really fixing any awkwardness. It's only the assumption that she did it herself that created the awkwardness it then seeks to fix, and then it accidentally creates another (admittedly slightly less) awkward phrasing.

So why should Rashi go to such great lengths (the pun wasn't originally intended, but it's a good one, isn't it) to cite this Midrash? Isn't his whole modus operandi to bring Peshat unless Peshat isn't adequate and the Pasuk calls for Derash?