Split attribution: Bereishis Rabbah 55:8 for Yosef personally harnessing the chariot; Rashi on Bereishis 46:29 for Yaakov’s reciting the Shema.
Sefaria — Bereishis Rabbah 55:8
וַיֶּאְסֹר יוֹסֵף מֶרְכַּבְתּוֹ — הוּא עַצְמוֹ אָסַר אֶת הַסּוּסִים. ... וְיַעֲקֹב הָיָה קוֹרֵא אֶת שְׁמַע.
English: Yosef himself harnessed the horses to the chariot; Yaakov was reciting the Shema.
On the daf / in context: These are distinct traditions and should not be attributed wholesale to Bereishis Rabbah 95. The chariot detail is preserved in Bereishis Rabbah 55:8, while the Shema explanation is associated with Rashi’s comment on the verse.
What Is Learned: Yosef’s personal effort expresses honor for his father; the Shema tradition explains Yaakov’s unusual response at the reunion.