Does serving a need (of another) after exiting the state of ihram for Umrah and before the intention to enter into the rituals of Hajj—after taking off the ihram garments and wearing regular clothes, then re-entering ihram for Hajj the next day—affect the validity of Hajj al-Tamattu?

The state of ihram (definition) for Hajj or Umrah (definition) is actualized by the intention, which is the resolve to begin the ritual. So, if you did not intend to enter into the rites of Hajj, then it was not actualized at all. What you did, reciting the talbiyah (definition) devoid of intention, does not make you in a state of ihram for Hajj. Consequently, nothing is binding upon you for removing your ihram garments after completing your Umrah.