There seem to be confusion with the questioner assuming Animal Sacrifice in the Vedas and consuming meat are one and the same. If the question is raised based on the Vedas, then Vedas should themselves answer it. Later literature can provide further definition, and justification and elaborate on its via examples, but can't be the source.
- Vedas have thousands of Yagnas. They all don't have animal sacrifice. The percentage of animal involvement in the aspect of their killing is very low. That means it's not a common practice. Vedas share all aspects of creation and existence. They talk about astrological, metaphysical, social, climatic, animals, human behavior, and more. Animal Killing is not excluded and not treated as a taboo. We will see with an example below.
- Animal Sacrifice doesn't always mean killing them. They can be donations of cattle as well. We will see examples of that as well.
- The same Veda that talks about Animal Sacrifice also talks about the evolution of human consciousness where the Vedas as the civilization to move away from such acts. We will see examples of those as well.
Rudra is the Lord and king of Sacrifice, Vājasaneyi Saṃhitā also describes a specific title called Pashunapati as the “Lord of Sacrifice of Pashu”; here Sacrifice of Cattle (Pashu) means to donate cattle as Dakshina during Yajñás like Rājasuya. The sacrifice of Cattle doesn’t always denote killing them, during the Vedic times human consciousness evolved towards sacrifices that do not require killing, for example:
न ता अर्वा रेणुककाटो अश्नुते न संस्कृतत्रमुप यन्ति ता अभि । उरुगायमभयं
तस्य ता अनु गावो मर्तस्य वि चरन्ति यज्वनः ॥ ४ “The war-horse never
catches them, they never go to the place of slaughter or sacrifice.
The Cows always roam without fear on the wide-spreading land of the
mortal worshiper.” ~Rig Veda 6.28.4
Please note: Rig Veda 6.28 is an entire hymn dedicated to cow.
“He replies, ‘The mother of the Rudras, the daughter of the Vasus, the
sister of the Âdityas, the navel of immortality. To the people who
understand me, I say, “Do not kill the guiltless cow, which is Aditi.”
I kill my sin,’—thus, if he chooses to have it killed. But if he
chooses to let it loose, he should say, ‘My sin has been killed. Om!
Let it loose! Let it eat grass!'” ~ Grihya-sutra of Paraskara Sukla Yajur Veda 1.3
So, what are these Paśūs? Taittirīya Saṃhitā 3.1.4 says “पशुपतिः पशू॒नां चतुष्पदामुत च द्विपदाम्” meaning “He is Paśūspati to Paśūs that are two-footed and four-footed”. Vājasaneyi Saṃhitā 184.108.40.206 Brahmāṇa 1.2 describes 5 types of Paśūs sought by Prajāpati – man, horse, bull, ram, and goat, similarly Atharva Veda 11.2.9 defines Paśūs as cows, horses, man, goats, and sheep. But if we take dictionary definition then Paśū means cattle or animals that are tamed, or a group of creatures, or creatures used as sacrificial gifts given to rishis, scholars, and Devas. But Paśu has a broader meaning, it includes all beings who are tied to the material world in the loop called Samsara. So Paśūs are those which are in bondage or tied/fetter, hence the term Pãsha. We can notice that they both have the same root (Paśū and Pãsha). Many western scholars limit Paśu to just farmed cattle, but that would make Mahadeva’s Paśupatiastra, the most powerful weapon given to Arjuna and Sri Krishna in Mahābhārata Itihāsa, was meant to kill cattle in the war.
Note: Just because the western translaters translated Pasu as animal sacrifice then you have seen the list above which includes human being as well. So, not everthing is mere killing.
How did the concept of Rudra diversify into Paśūspati? Across Vedas, there are numerous instances wherein Rudra emerged and took ownership toward all Bhutas (beings) and Paśūs. Two events take high precedence, one is from Kausitaki Shankhyayana Brāhmaṇa Kanda 6.1-9 and Śatapatha Brāhmaṇa Kanda 6 Brahmāṇa 3 wherein Rudra expands into Paśūspati title by encompassing all flora/fauna. Second is the famous event of Tripurasuras from Taittirīya Saṃhitā 6.2.3 wherein Rudra claims this title. Other Brahmāṇa sections like the story of Bhutavān AB3.3.33 do talk about many exploits wherein Rudra emerges and claims this title. Since Rudra expanded as Paśūspati and claims ownership of their sacrifice, one has to take Rudra’s permission before such an act.
भव राजन् यजमानाय मृड पशूनां हि पशुपतिर् बभूथ । यः श्रद्दधाति सन्ति
देवा इति चतुष्पदे द्विपदे ऽस्य मृड ॥11.2.28 Oh Bhava (भव ) you are the
ruler/king (राजन् ) be gracious/favorable, pardon/spare (मृड ) towards
the one hosting the Yagna (यजमानाय ) especially which donating cattle
as sacrifice (पशूनां ) especially you are the one to reach (बभूथ )
being the lord of cattle (पशुपतिर् )| those who (यः ) have faith in
the divine (श्रद्दधा श्रद्-दधान देवा ) view graciously (सन्ति) thus on
(इति ) our four legged (चतुष्पदे ) and two legged (द्विपदे ) Please
note: Yajamāna = the host of the ceremony, Adhvaryu = the one making
oblations ~ Atharva Veda 11.2.28
Note: By now the readers should understand that eating meat every week or for parties or for taste and habit has NOTHING to do with Vedas justifying it. We human beings are not equipped to hunt with our own hands or run/chase behind prey and kill it and eat it raw. Our own primates (chimpanzees) don't eat meat. Out Digestive tract and our teeth are not equipped to kill and tear meat. Today we want someone else to kill it with a weapon/tool, clean it, cut it, cook it, and put it in front on our table with spices so that we have no notion or guilt that the content of our place is actually an animal. Vedas do not talk about any of this. So using Vedas to justify eating meat is a sick way to clear our consciousness
Do Vedas use the term Pãsha as the bonds that tie us to material worlds? Does Rudra free the Paśūs from these Pãsha? Of course, let’s examine the Upaniṣhads and link them back to Vedic Saṃhitās.
देवं ज्ञात्वा सर्व-पाशापहानिः भवति । क्षीणैः क्लेशैः जन्ममृत्युप्रहाणिः। तस्य अभिध्यानात् देहभेदे तृतीयं विश्वैश्वर्यं च सः केवलः आप्तकामः च भवति ॥ 1.11
With the knowledge (ज्ञात्व) of the divine (देवं) all (सर्व) bondages/fetters (पाशा) looses (हानि) its existiance (भवति) | With the depletion (क्षीणै) of disturbances in mind (क्लेशैः) and knowledge all the cycle of birth (जन्म) and death (मृत्यु) ceases ((हानि)) | Therefore (तस्य) through meditative (ध्याना) states overcome bodily desires and changes (देह-भेदे) and enter into the Turiya (तृतीयं) state (third state) and (च) in turn realizes that/His (सः) Sovereign Authority of entirety (विश्वैश्वर्यं). In this state one alone (केवलः) exists without a second, and all notion of self and its desires reach completion (आप्तकामः).
घृतात् परं मण्डम् इव अतिसूक्ष्मं शिवं सर्वभूतेषु गूढं ज्ञात्वा विश्वस्य एकं परिवेष्टितारं देवं ज्ञात्वा सर्वपाशैः मुच्यते॥4.16
like the cream (मण्डम् ) above (परं) the clarified butter (घृतात्) the hidden within (गूढम् ) in all beings/existences (सर्व+भूतेषु ), Knowing (ज्ञात्वा ) him who is exceedingly (अति) subtle (सूक्ष्मं ) who is Sivam/the auspecious one (शिवं ) in all, knowing (ज्ञात्वा ) that divine (देवं ) who along (एकं ) encompasses all (विश्वस्य परिवेष्टितारं ), one is released (मुच्यते) from every bondage (सर्व+पाशैः ).
Svetasvatara Upaniṣhad 1.11/ 4.16
This is not the only statement, Svetasvatara Upaniṣhad 1.8, 4.15-16, 6.13 and Taittirīya Āraṇyaka 10.47.74 talk about releasing oneself from Pãsha in clear details.