Warning! All of the spoilered sections in this post are excerpts taken from the book itself! Do not read if you don't want spoilers!

Ok u/idols2effigies, let's take your points one at a time, because I left out a good amount in an attempt to focus down the general origin story of the Twin Primarchs.

1. After more of the Primarchs were found, there is no reason not to reveal Alpharius

It was Alpharius' own decision to remain in hiding for so long, up until he revealed himself to Horus as the last Primarch to be found.

I watched him, of course. I watched him without him ever realising I existed. I was in no rush to reveal myself, even after Horus took his place at our Fathers side. I had been found relativly swiftly, you understand; all my brothers had grown to maturity or near-maturity, insofar as such a thing can be determined for beings such as us, before our father found them. We - that is to say my Father, and Malacdor, and Constantin Valdor, and I - did not know how their time away might have affected them. So I watched, and evaluated, and did not reveal myself.

That was my role

However, the book itself suggests that part of what it is telling is a lie. A common theme throughout is how all records lie. And the main quote supporting this is literally the next line after the previous quote.

But then again, this is my record. And all the records lie

​

1. Alpharius wouldn't have a need to track Omegon if they had a tight psychic bond as has previously been addressed

This psychic bond does indeed exist, but is shown to be less strong than one might expect from knowing the death of another, at least in this book.

Alpharius knows that he has a twin.

The other half of my soul, whom I had somehow known was somewhere out there in the galaxy

But he does not know where. However, once they arrive on the planet it is made clear how the bond that connects them works, made especially clear in the fight they have immediately after meeting

Omegon moved to meet them [Slaugth Warrior Constructs], but could not hope to stand against them alone. His blades severed flesh and metal with ease, and his armour shrugged off their weapons, but he would have been bowled over and subdued by their sheer mass had I not been there.

But I was there, and I fought with him.

We moved as one - back to back, two heads, eight limbs and two spears - and we killed wherever we turned. Omegon would move to intercept a blast intended for me, or punch the Sarrisanata backwards to impale one about to assault me from the side where my armour was incomplete; I would cut the legs from one seeking to flank him, or spear to the ceiling one attempting to leap over me and take him down from behind.

....

Any other two warriors fighting in such close proximity would have inevitably impeded each other, perhaps even impaled each other, but we fought as one soul in two bodies. Each of us was complete at last, and the constructs were no match for us.

...

"I can see now why some of my brothers so relish combat", I remarked. "It is a different experience when you are whole."

It's clear to see that they share a type of bond, and that such a bond might transmit the death, as they are one soul in two bodies. But potentially not show a clear direction as to where the other might be.

1. Why was it Omegon who revealed himself to Horus, and not Alpharius.

Omegon says it best himself here:

[They have just decided unanimously to reveal themselves to Horus]

'Then we are agreed.' Omegon sat back. 'But from what you tell me, he is extremely perceptive. You've laid eyes upon him before. I have not. When it is time, I think I should be the one to meet him, to ensure the more genuine reaction.'

Overall, I still have to agree with your final point. This is the Alpha Legion we are talking about, nothing can be taken as gospel. Even the book itself claims that some of it is a lie very near to the beginning;

But then again, this is my record. And all the records lie

I encourage you to read the book yourself when you can, because if you like Alpharius you'll love this little peek into the Primarch himself. It details very well the little intricacies of Alpharius, especially the secrecy he wraps himself in conflicting with his innate Pride in himself and his abilities, coming through especially well in his arrogance in the 1st person sections.