I have read your post, I have thought about it, and I agree with most of what you say. For the sake of keeping it short, however much I enjoyed our lengthy posts, I'll refrain from writing too much. As you say, let's keep it shorter!
I'll just pick a few points of yours and answer directly to them, because those warrant a bit of diving deeper into:
However what I deeply dislike here is that it make a lot of empire fully soulless.
Well, is being soulless such a bad thing for a materialist? To be honest, if you reject the notion of a soul as superstition, if you don't care if your spiritual
you can project itself into another dimension, if for you the material realm is all that matters, why would the absence of a soul bother you? I think it wouldn't, not one bit.
You yourself ask "because if you can do everything a soulfull being can at perfection, without needing a soul, what is the point of having a soul in the first place ?"; the materialist would confidently say that there is no point in having a soul. The spiritualist would accuse you of being a soulless machine. And that is the core of the dislike between both, a fundamentally different view on the necessity of having a soul.
To a materialist it doesn't matter. Just as individualism doesn't matter to a hive. Individualist empires dislike hive minds, but is it really a bad thing to be a hive? Or not being a hive? Does the absence of individualism constitute being not fully alive? Wouldn't we rather say that life takes different forms, and being a machine without a soul is also life, albeit without a spiritual presence?
Look at the soul as a modus of being, one attribute among many. What does it matter (literally) to have a soul? To take the example of Roy, who as a human-like robot acted as a human would; do you need a soul, if you have a proverbial heart?
Now let say soul does indeed exist in stellaris but is not what spiritualist think it is. The soul can exist without shroud presence.
I wouldn't tie the Shroud to the soul. It's not a necessary connection, as is evident with the Zroni origin story. They were the first to open that dimension (which they found in a very raw and empty state), enter it physically, and use their psionic power to shape it. The shroud entities are a result of Zroni feelings coming to life, so in a way new spiritual presences, god-like presences, emerged.
But the Zroni are definitely not the first to have a soul, and non-psionic bio empires also don't touch the Shroud at all. So the Shroud is not a measure for the soul, although it seems that having a trained(!) soul/spirit would be a requirement to interact with it in a meaningful way.
I'd say the common soul, in its most basic iteration, is a metaphysical presence that defines you beyond the atoms and molecules that constitute your body. And that metaphysical presence is able to influence this dimension as well as others (such as the Shroud), if you train it to do so.
It's not easy to pinpoint what it is, because if we take away the body, what's left of a person? If there is absolutely no body, no sensory input, wouldn't there be souls without a body floating around? We could even ask, can souls emerge out of nothing without a body, without being born alongside a physical biological body? I'd say there is an initial connection between souls and biological bodies, but both don't have to remain connected.
That's why I would say the soul is merely your spiritual presence, starting out as a union of body and mind, and it's up to you what you make of it. If you're going down the psionic path, you project/imprint attributes of your body (such as memories, your "will", sensory abilities) onto that presence. That allows you to ultimately ditch your biological body and continue existing as you. But you can also disregard it and project your entire body and brain on a machine frame.
The game, and to some degree even our reality, is based on evolution. I like to think that the soul can also evolve. It's actually a Buddhist way of looking at it, because according to its teachings, your soul ditches bodies on its way to spiritual fullfillment. It learns, evolves, takes memories with it, until it finally frees itself from the cycle of reincarnation by discarding all attachment to the physical realm.
My conclusion therefore is, that the presence or absence of a soul does or doesn't matter, depending on who you ask. That means that when it comes to psionic robots, we're talking about spiritualists, who very much mind the presence of a soul. So the notion of a soul-infused psionic robot makes sense for them, because they care that it has a soul.
Oh, and btw, there are new portraits for Psionic phenotypes, if you haven't seen them yet. So far none of them seem robotic:
