The first rule of 8 Garudhammas

In this other question (which was closed as a duplicate of this one), the OP wrote,

One day during a Dhamma deshana I heard a thero chanting that these rules were added later into Buddhism, by Brahmins, to show that Lord Buddha didn't give equal right to women, which was a practice in their community. Can someone please clarify this to me?

I note that Wikipedia references papers which claim to show that the rules were added later:

However, Bhikkunī Kusuma in her article "Inaccuracies in Buddhist Women's History" has pointed out a number of inaccuracies in the ways the Eight Garudhammas have been recorded in the Pali Canon and its commentaries.[3] And others point out the plethora of textual problems with the position for garudhammas.[15] Tathaaloka Bhikkhuni published evidence that the Eight Garundhammas are non-historical.[5]

So, I don't know but these references suggest that the Thero isn't the only person with this belief.

See also this answer, which gives details of some modern-day scholars' attempts to demonstrate this kind of thing (i.e. to demonstrate that it's a later addition) on a different but similar subject.

How important this rule is, is best shown by the first Bhikkhuni herself, as she asked the Buddha to repeal the rule, because it's hard, and thereby was not only rejected but also rebuked for her respectlessness.

Bhikkhunīs

Just as a clan in which there are many women and few men is easily plundered by robbers and thieves, in the same way, in whatever doctrine and discipline women get to go forth, the holy life does not last long... Just as a man might make an embankment in advance around a great reservoir to keep the waters from overflowing, in the same way I have set forth in advance the eight rules of respect for bhikkhunīs that they are not to transgress as long as they live." — Cv.X.1

[...] The analogy of the clan predominantly female shows that, in the Buddha's eyes, the survival of the holy life required a Community predominantly male. That was why he delayed granting Acceptance to his aunt, so that she would be willing to accept the eight rules of respect; that was why the requirements for Acceptance in the Bhikkhunī Saṅgha were more difficult and complicated than the requirements for Acceptance in the Bhikkhu Saṅgha; and that was why many of the rules governing relationships between the two Communities favored the bhikkhus over the bhikkhunīs.

The bhikkhunīs did not accept this situation docilely. Soon after vowing to adhere to the eight rules of respect for the rest of her life, Mahāpajāpatī Gotamī requested that the bhikkhunīs be relieved of the most onerous one — the first. The fact that she was asking to renege on her word to the Buddha doomed the request to failure.

My person will not explain in detail, because "political correctness" and doing a favour for personal gain are today much more dominant, and people neither tend to faith and truth, nor to what is required for long term benefit, they generally have actually no respect in regard of the Jewels and put their own developed values above it: do not go for refuge = putting one self under an authority.

For one who wishes to seriously understand, it will not easy, but will, but one find the answer briefly given to an extent that will satisfy one's last doubts.

Others may even see in this very lifetime the reason, but then, if the Savaka Sangha is finally extinguished, it will be just a shock: "oh..." that would left behind.

(Note: you may try to sell this gift of Dhamma or even use it for your personal wordily gain! This will show you the truth of it's content.)

There's interesting information provided in this website, which provides some (presumably) anecdotal justification for the rule.

The first story from the Sarvāstivāda Vinaya Suttavibhaṅga is as follows:

(Thullananda is a Bhikkhuni)

The lay woman went to Thullanandā and said: ‘Are you aware that was the Elder Mahākassapa, the Buddha’s great disciple, who is greatly revered by the deities as a virtuous field of merit? If you were to rise and greet him, what harm would come of that?’

Thullanandā said: ‘Mahākassapa was originally practicing another religion, [i.e.] Brahmanism. You greatly reverence that, but I do not respect it.’

The lay woman was annoyed and scolded: ‘These bhikkhunis say, “If you do what is good you will get merit”, but when they see bhikkhus coming they do not rise, as if they were women from another religion.’

When the bhikkhunis of few wishes, contented, keepers of ascetic practices heard about this they were not pleased. They went to the Buddha and told him everything. For that reason the Buddha summoned the two-fold Sangha together.

Knowing, he asked: ‘Is it true that you did that thing, or not?’

She answered: ‘It is true, Blessed One.’

The Buddha for this reason in many ways scolded: ‘How can this bhikkhuni see a monk coming and not rise?’ Having in many ways scolded for that reason, he said to the bhikkhus: ‘For the sake of ten benefits, I lay down this precept for bhikkhunis. From today onwards that precept should be taught:

‘Should a bhikkhuni, seeing a bhikkhu coming, not rise, this is an offense of pācittiya.’

The second story from the Vinaya of the Mahīśāsakas:

Now at that time bhikkhunis did not bow to monks, did not greet them, did not receive them, did not invite them to a seat. The bhikkhus were annoyed, and did not return to teach. Then the bhikkhunis were foolish, without knowledge, and not able to train in the precepts. The senior bhikkhunis saw this, looked down on it, and scolded in many ways. The matter was therefore told to the Buddha. For that reason the Buddha summoned together the two-fold Sangha.

He asked the bhikkhunis: ‘Is this true or not?’

They replied: ‘It is true, Blessed One.’

The Buddha in many ways scolded them: ‘Did I not already teach the eight garudhammas as suitable etiquette regarding bhikkhus? From today onwards, that precept should be thus recited:

‘Should a bhikkhuni, seeing a bhikkhu, not rise up, bow down, and invite him to a seat, this is an offense of pācittiya.’

For trainees and novices, it is an offense of wrong-doing. If sick, if previously there is anger and suspicion, with no shared speech [recitation?], there is no offense.’

And Pācittiya means:

‘Pācittiya’ means: burn, boil, smear, obstruct. If not confessed, it will obstruct the path. This is the offense: if a bhikkhuni sees a bhikkhu and does not rise, this is a pācittiya; straightaway seeing and not rising, straightaway at that point there is pācittiya.