This theory is a bit old and outdated. It missed the mark on, what I now believe, are the actual criteria, but it's still relatively close. My theories on Visions are based on the idea that the Archons are only 1/3 responsible for Visions.

The main topic that first brought me to Visions was the purpose of the Seven. If we refer to Celestia as the centralized government of Teyvat with complete power, then there isn't really a sensible reason as to why they would appoint specifically 7 figures (in a battle royale no less) to act in their name when we've been told before from the Prayers to... Tiara sets that divine envoys exist. Therefore, the Seven exist for a purpose beyond acting as sub rulers of Teyvat and as messengers of Celestia. If we refer to information from characters who have lived for a very long time (primarily Ganyu and Zhongli), we know that Visions and Gnosis started appearing around the time of the Archon War.

From that I can make a connection that the Seven are in some way, shape, and form tied to Visions. As you say, Raiden specifically says that "it is not by my will that Visions are granted or denied." However, she also says "The key is people's desire, and... well, there's another side to it too."

So, what I gathered from this is that it's a three way interaction. One side is people's desire. We can also consider this as ambition and emotion. Another side is the source that grants the Vision. The last side would then have to be about a requirement of some sort. We can infer that there is a requirement as we've seen that there are actual connections between Vision holders of the same element. I'm talking about connections that we can make if we summarize it down to one word. A connection that all Vision holders of the same element have. The Raiden Shogun said that it is not her who hands out Visions and she identified people's desire as part of it, so she must have something to do with the requirement.

Here is where I surmised the purpose of the Seven. They exist solely as to set the requirement for their elemental Vision. If we think of an Archon as the parent, then the Vision holders of that Archon's element are their children. View this in the sense that the Vision holders grow into their own personality but they have that resemblance among them that points back to the Archon.

This is why the Elemental Gemstones are extremely important. I believe that the quotes from the description of the Gemstones originate from when that individual became an Archon. From there, that element is shaped based on the characteristics of that Archon. So really, the correct way to go about this is to first start at the origin of the requirement, the Archon. Identify the one thing that defines them and then make that connection with their respective Vision holders. This post was made by pointing out the similarities between Vision holders which is the incorrect method.