|ח וַיּוֹצִאֵנוּ ה, מִמִּצְרַיִם, בְּיָד חֲזָקָה וּבִזְרֹעַ נְטוּיָה, וּבְמֹרָא גָּדֹל--וּבְאֹתוֹת, וּבְמֹפְתִים.||8 And the LORD brought us forth out of Egypt with a mighty hand, and with an outstretched arm, and with great terribleness, and with signs, and with wonders.|
And in particular on וַיּוֹצִאֵנוּ ה, מִמִּצְרַיִם.
Yes, there is repetition of Hashem here. But the verb vayar does not make use of it. And so too the next pasuk (vayotzi'einu) could have omitted explicit mention of Hashem. And so too, in the next pasuk:
וַיְבִאֵנוּ, אֶל-הַמָּקוֹם הַזֶּה; וַיִּתֶּן-לָנוּ אֶת-הָאָרֶץ הַזֹּאת, אֶרֶץ זָבַת חָלָב וּדְבָשׁ.
No explicit mention of Hashem is made. So this might be what prompts the derasha.
In terms of simple peshat in this pasuk, when read in full, indeed, the basic idea is that Hashem brings forth His full might to take the Jews out of Egypt. So that might be what prompts the derasha. Alternatively, the ani Hashem and other features of the prooftext prompt the derasha local to Arami Oved Avi.
The prooftext / manifestation of this idea is from makkat bechorot, the last of the makkos:
|יב וְעָבַרְתִּי בְאֶרֶץ-מִצְרַיִם, בַּלַּיְלָה הַזֶּה, וְהִכֵּיתִי כָל-בְּכוֹר בְּאֶרֶץ מִצְרַיִם, מֵאָדָם וְעַד-בְּהֵמָה; וּבְכָל-אֱלֹהֵי מִצְרַיִם אֶעֱשֶׂה שְׁפָטִים, אֲנִי ה.||12 For I will go through the land of Egypt in that night, and will smite all the first-born in the land of Egypt, both man and beast; and against all the gods of Egypt I will execute judgments: I am the LORD.|
Wasn't Moshe a shaliach? Well, perhaps in the other makkos, when he is commanded to take some practical or symbolic action to bring about the plague. But at the very end, with makkat bechorot, there is no such action. Hashem acts directly. And so is the sense, and the flavor, of the entire perek in sefer Shemot.
We might have been tempted to say otherwise, in light of the very last few pesukim of the Torah:
All the otot, mofetim, yad hachazaka, and mora hagadol, all mentioned in Arami Oved Avi in this pasuk, are wrought by Moshe, as messenger of Hashem. Still, makat bechorot is not of this nature. And this could be the intent of the stress in this derasha.