Here is my take in Layman's terms in conjunction with Nazz's explanation.

You will need the power of the Aedra and the Daedra in order to create a new world, in this case the blood of both was used. You will need a conduit of energy powerful enough to envoke the magical properties of the blood in order to create a new world. The world is bound in Magic, its very existence depends upon it. Nirn was bound using the magic of Magnus and so the world continues. In this case a Great Welkynd stone was used in place of Magnus.

This process was explained in detail in the Mysterium Xarxes.

Not only would one need these items to create the world but one would need to understand what keeps the world together. Nirn is kept together by the light of the Aetherius (spelling?). This keeps the Void from crushing Creation. The Amulet of Kings is a suitable replacement as it contains a part of then Aether, the very lifeforce of Akatosh.

Mankar states that since the heart of Lorkhan is the heart of the world and to destroy the heart is to destroy the world ( something we learn in MW), he believes that the world itself belongs to the Daedra. Also, by his assumption he believes that Paradise would be indestructible as there is no "heart of the world" but Mankar overlooked the fact that Lorkhan and Akatosh are both remnants of the same entity.

Mankar Camoran paired with the Amulet of Kings became the "heart of the world". By removing him from the Amulet of Kings, Paradise was destroyed.

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As for the OP's original inquiry: Anyone with sufficient knowledge of the Aedra, the Daedra, Magic and how it all ties together, could accomplish anything using the artifacts. The artifacts themselves are raw magical power manifested as an object on the mortal plane. Seperate the power from its manifestation and you have the building blocks of Creation but without significant magic energy such as a Great Welkynd stone or a very powerful wizard (Magnus, Fyr) this process would be next to impossible.

I could be wrong.....

And, on a side note, if the Blood of the Divines is Sooooo hard to find, and there's only one instance of it (Besides maybe the later-introduced artifacts of the crusader) is Tiber Septims armor, how did Mankar Cameron get into the Paradise?

(I think Proweler may have mentioned this in a different wording...)

I suspect that because Mankar Camoran was bound to his Paradise, he didn't need the artifacts to open a portal.

He visited the version of the nerevarine that was a little bit on the stabby side with those avatars of the Divines in morrowind.

I always felt gypped when it game to blood of a divine. Marin & Co. are all too eager to insist that he is the dragonborn and the descendant of the divine Tiber Septim, but as soon as it comes to spilling a bit of blood suddenly they'd rather some flakes of rust from an ancient and unrelated (in the most literal sense) suit of armour.

It may not have actually worked, we'll never know or sure now, but it made a whole lot more sense to me.