Does acknowledging our ego and not letting it win over us is itself ego?

Param Pujya Dada Bhagwan says, “Insistence is a photo of the ego!”
The ego of any person can be understood from his insistence. And liberation can never be attained through insistence of opinion; it is only when one becomes free from all prejudices, partialities and non-insistence that one becomes successful. Thus, by not letting your ego win, you are taking the right step in the direction of being successful, in the direction of being liberated from ego.

Insistence Endangers Dissent or Conflict, and That Comes in the Way of Liberation.
So as per me, giving up the argument is the most right thing to do! Further, you say, you acknowledge that it is your ego’s play when you know they are right, but still you argue. This again is the best thing to do!

Now coming to your main question, “Is not letting the ego to win also an ego?”
For this, would you like to know what really is ego? Let’s understand that in the light of the Enlightened One’s teachings.

First of all, let’s find this out...
What you believe to be right, how far is it right?
If someone asks you, are you yourself John (you may insert your own name here), or is your name John - what would you answer?
“It is my name.”
Then, who are you? If John is your name, then who are you? If you say, "These are my eye-glasses", then you and the glasses are separate, right? On the same lines, if you say, “John is my name”, you and John are separate. Yet, people insist, "No, I am John.” And they believe, “I am right!” ï

What really is ego?
Now, let’s find that out as well...
John is just a means of identification. From your very childhood, people kept calling you John and you have come to believe, "I am John." You believe this name to be you. In reality, you are not that. But even then, you insist you are 'John' because everyone tells you so.

Since you do not know who you really are, you believe yourself to be the name that you have been given. This has had a very powerful psychological effect on you. This effect is so deeply ingrained within you, that you strongly believe that you are really John. This belief is wrong. And this very wrong belief is called ego!

At the moment, you always identify with the ‘My’
What are all the things that fall under ‘My’? Let’s understand...
Home, all the things inside home, mind, all the opinions and the viewpoints in mind fall under ‘I’ or ‘My’? Do you say, “My mind” or “I am mind”? And the egoism? Do you say, “My egoism” or do you say, “I am egoism”? You say, “My egoism.” So even egoism is not what you are. So who are you?

Who am I?
That ‘I’ is precisely what you are! And that is the ‘I’ that you need to realize (Self-Realization).
Will you not have to investigate who you really are? For how long can we go on being in the dark, unaware of our real Self? Do you not think it is ignorance to not investigate your true identity? Until you realize who you really are, everything proves to be wrong and incorrect. Without knowing the answer to this one vital question, “Who am I?”, we only complicate our life further by actively participating in this relative world. This is how confusion and puzzles arise in the relative world.

Only when you realize the answer to the question ‘Who am I?’ does your wrong belief stop; does your ego stop, because as explained above, this wrong belief itself is called ego.

The belief of ‘I am John’ is ahamkar (ego)
To impose ‘I’ where ‘I’ does not belong is ahamkar (ego). Whenever you superimpose ‘I’, where ‘I’ does not belong, that is a wrong belief. You have to get rid of this wrong belief because this blunder is the root cause of all suffering. This suffering comes in the form of inner restlessness, misery and discontentment. At the moment, you have so many worries caused by your false belief of, “I am John.” This false imposition on the Self is ignorance, and brings forth suffering.

How will you get rid of the ego?
When you identify ‘I’ with your Real Self (its proper place), then it is not ego. It means if the ‘I’ comes out of its incorrectly, positioned place (John) and arrives in its Real place (the Pure Soul), then the ego is gone. Therefore you do not have to get rid of ‘I’; ‘I’ just needs to be put in its correct location.

“I” is God (the Self) and “my” is the illusory attachment (maya). “I” is real.
Only a Gnani Purush (Self-Realized one) can make this separation between ‘I’ and ‘my’ happen. Gnani Purush means the Enlightened One! He has the divine power to separate the Self (“I”) and the non-Self (“My”), thus leading to Spiritual Enlightenment.

Just like only a goldsmith can separate the gold and the copper because he knows the qualities of both of these elements; similarly, a Gnani Purush knows the qualities of the Self (Pure Soul) and the non-Self (everything other than the Soul that we identify with). Hence he can separate the two and one is able to understand the Self. Pure Soul is the Self. That’s what you really are.

An ardent desire to know the Absolute Truth and insistence of our opinions are contradictory.
God has asserted that one should not fall a prey to any clinging insistence within us, except in seeking the Gnani, at whose lotus feet lies Liberation. The grace and blessings of the Gnani would place you right on the threshold of Liberation without any strain or pain on your part. So, go to Gnani and get rid of your ego soon!

To know more: https://www.dadabhagwan.org/path-to-happiness/spiritual-science/what-is-ego-egoism/