This reflects the Mishna and gemara in Pesachim. The Mishna stated:
מתחיל בגנות ומסיים בשבח:And the gemara upon that gave a dispute between Rav and Shmuel. Thus,
מאי בגנות רב אמר מתחלה עובדי עבודה זרה היו אבותינו [ושמואל] אמר עבדים היינוThis explanation of Rav is a topic, rather than a pasuk. The Talmud Yerushalmi, masechet Pesachim, gives us the actual Pesukim which Rav intended:
דף ע, ב פרק י הלכה ה גמרא רב אמר מתחילה צריך להתחיל (יהושוע כד) בעבר הנהר ישבו אותיכם וגו'. ואקח את אביכם את אברהם מעבר הנהר וגו'.ש
I think someone (credit upon request) suggested to me that Rav never meant to say the words mitchila. Rather, he says in Yerushalmi that in the beginning, he should start as follows, namely with the following pasuk. And this was (mis-)interpreted as that he needs to begin with mitchila. And this is then taken from the Bavli, leading off with what Rav defines as the genus. This seems plausible.
We know the genus, but what is the shevach? I don't see the gemara explicitly mention it, in Rav's name. The Haggadah does give an answer -- that now, we are brought close before Hashem. This is the opposite of the genus, and so it makes sense. But it could also be going into the land, or being redeemed from Egypt. This is, after all, supposed to be a sippur yetzias Mitzrayim. Is Rav's recommendation supposed to be a self-contained beginning and end?
If we start with idolatry and end with service of Hashem, this is indeed the topic of the entire last perek of sefer Yehoshua. We can start with:
|ב וַיֹּאמֶר יְהוֹשֻׁעַ אֶל-כָּל-הָעָם, כֹּה-אָמַר יְהוָה אֱלֹהֵי יִשְׂרָאֵל, בְּעֵבֶר הַנָּהָר יָשְׁבוּ אֲבוֹתֵיכֶם מֵעוֹלָם, תֶּרַח אֲבִי אַבְרָהָם וַאֲבִי נָחוֹר; וַיַּעַבְדוּ, אֱלֹהִים אֲחֵרִים.||2 And Joshua said unto all the people: 'Thus saith the LORD, the God of Israel: Your fathers dwelt of old time beyond the River, even Terah, the father of Abraham, and the father of Nahor; and they served other gods.|
where the focus is, admittedly, that they dwelt be'ever hanahar, but more importantly, that they served other gods. Yehoshua asks whether now the Israelites will commit themselves solely to Hashem, saying:
And their response is:
And thus, because of the redemption from Egypt (pasuk 17), we have been brought before Hashem, and accept him as our God, as so we serve him, for He is our God. Perhaps, then, the intent of Rav was to read the entire perek.
The pesukim cited in Yerushalmi are only two -- the first two, be'ever hanahar and va'ekach. But in our Haggadah, the citation of the first pasuk begins a bit earlier, from vayomer Yehoshua, and it cites three pesukim:
Thus, there is also the va'eten. What gives? Perhaps there was another tradition, either that we have and I don't know about, or which did not come down to us.
When the Yerushalmi cited those two pesukim, was it a self-contained beginning and end? (Just as Shmuel's avadim hayinu has immediately after it Hashem taking us out.) If so, vaekach would be Hashem taking Avraham before him. Thus, קרבנו לפני המקום.
When the Haggadah cites the third and fourth pasuk, how exactly is this ending in shevach? Presumably -- and this would answer many of the difficulties -- this is all part of the mitchila. We needed to transition to the avdus in Mitzrayim. The Mishna in Pesachim says to start with genus and end with shevach, and that he is doresh from Arami Oved Avi. If this is a running narrative, we get up to Egypt, and at that point begin the derasha of Arami Oved Avi. Shmuel gets there, to Egypt, immediately. Rav does not, at least not with his two pesukim. And so, rather than reading almost all the pesukim in the last perek of Yehoshua; and rather than reading only the first two, the Haggadah has us read a third and fourth pasuk, which takes us down into Egypt. And from there, we can follow the instructions of the Mishna about being doresh Arami Oved Avi.
Alternatively, the Shevach is somehow coming down to Egypt, because there we will develop into a nation, in accordance with Hashem's promise to Avraham at the Bris Bein Habesarim. And the Baruch Shomer Havtachaso might develop this idea.