The mantra you mentioned isn't actually part of the Rudram. The Rudram and Chamakam constitute Prapathaka 5 and Prapathaka 7 of the Fourth Kanda of the Taittiriya Samhita of the Yajur Veda. In contrast, the mantra you're looking for is from Prapathaka 3 of the Third Kanda of the Taittiriya Samhita:
vāyúr hiṃkartā́gníḥ prastotā́ prajā́patiḥ sā́ma bŕ̥haspátir udgātā́ víśve devā́ upagātā́ro marútaḥ pratihartā́ra índro nidʰánaṃ té devā́ḥ prāṇabʰŕ̥taḥ prāṇám máyi dadʰatu |
etád vái sárvam adʰvaryúr upākurvánn udgātŕ̥bʰya upā́karoti té devā́ḥ prāṇabʰŕ̥taḥ prāṇám máyi dadʰatv íty āhaitád evá sárvam ātmán dʰatte |
íḍā devahū́r mánur yajñanī́s |
bŕ̥haspátir uktʰāmadā́ni śaṁsiṣat |
víśve devā́ḥ ||
pŕ̥tʰivi mātar mā́ mā hiṁsīs |
mádʰu maniṣye mádʰu janiṣye mádʰu vakṣyāmi mádʰu vadiṣyāmi mádʰumatīṃ devébʰyo vā́cam udyāsaṁ śuśrūṣéṇyām manuṣyèbʰyas
tám mā devā́ avantu śobʰā́yai pitáro 'nu madantu ||
a The maker of the sound 'Him' is Vayu, the Prastotr is Agni, the
Saman is Prajapati, the Udgatr is Brhaspati, the subordinate singers
are the All-gods, the Pratihartrs are the Maruts, the finale is Indra;
may these gods who support breath bestow breath upon me.
b All this the Adhvaryu, as he begins, begins for the Udgatrs; 'May
these gods who support breath bestow breath upon me', he says; verily
he bestows all this on himself.
c May Ida who summoneth the gods, Manu who leadeth the sacrifice,
d May Brhaspati recite the hymns and acclamations.
e The All-gods are reciters of the hymns.
f O earth mother, do not harm me.
g Of honey shall I think, honey shall I produce, honey shall I
proclaim, honey shall I speak, may I utter speech full of honey for
the gods, and acceptable to men.
h May the gods aid me to radiance, may the Pitrs rejoice in me.
As you can see here, this mantra is one of "[t]he Mantras for beginning the Stotra and the Pratigara". Let me explain.
As I discuss in this answer, Yagnas generally involved three priests, a Hotri who would utter Rig Veda mantras in their original form heard from the gods, an Advaryu who would utter Yajur Veda mantras while tending to the details of the Yagna, and an Udgatri who would sing songs from the Sama Veda. (Later a "brahmana" for the Atharvana Veda was added.) Now as described in this chapter of the Shatapatha Brahmana of the Yajur Veda, there was a certain part of the Soma Yagna in which the Udgatri would utter a hymn called a Stotra, and in response the Hotri would utter a hymn called a Shastra, and in response to that the Adhvaryu would utter a hymn called a Pratigara. This mantra was one of the mantras used to initiate that part of the Yagna.
The use of part of the mantra is also described in this excerpt from the Shankhyana Shrauta Sutras:
In this manner having summoned (the deities) he sits down with his knees raised and mutters, after having touched the earth with the span of his right hand: "May I not be separated from this firm standing, O Mother Earth. Hurt me not, scorch me not. I shall think what is sweet, wish what is sweet, engender what is sweet. I shall today utter words sweet to Gods, dear to men. Here do I by means of the fifteen-fold thunderbolt drive away my spiteful rival."
So to sum up, this mantra has no connection in the Vedas to the recital of the Rudram or Chamakam. People today may just use this as a Shanti mantra for the Chamakam because it's a mantra asking for blessings from all the gods. It would be similar to how the "gananam" mantra is often used nowadays as an Ahavaniya mantra in Yagnas, just because you're supposed to pray to Ganesha in the beginning of rituals. (Except the gananam mantra is not actually addressed to Ganesha, as I discuss here.)