Can you create a harmonious color wheel?
Probably, depending on how functional you need it to be.
A conventional 12-hour color wheel is really just a block diagram arranged a circle. It uses the placement geometry of the 12 divisions to show relationships between the colors. The spectral spacing of those colors around the wheel is distorted to force primary colors and complements into their locations on the diagram.
The gradients between the primary colors on the wheel are just artifacts of this process. The color distances around the wheel look like whatever they look like as a result. The wheel, when used in the conventional way, wasn't intended as a tool for accurately representing the color distribution of the spectrum, let alone identifying harmonious colors.
If the color wheel is built on true primary colors, it can perform a number of functions. Complementary colors are across from each other, with the black or white point in the middle. The primary colors can be used to create pure desired colors. You can get a sense of whether any secondary colors might be reproduced poorly by those base colors. etc.
You can use the format of the color wheel to plot other combinations of base colors that aren't true primaries and see how they work together, but it would have more limited use. It couldn't perform at least some of those functions.
If you didn't care about most of those functions, you could just use the format to show a circular progression of colors for some other purpose, like a harmonious color scheme. (Depending on the scheme, some of the wheel's features could function.)
But consider whether the impetus for this effort is focused in the right place. Is the purpose to create a harmonious scheme of 12 colors, or a system for producing harmonious colors. I'll come back to that later.
Are the ideas in the question on-track to create a harmonious color wheel?
Again, that depends on how functional you need the color wheel to be. Some input on specific points:
Limit of two base colors The question suggests starting with two base colors and then using various mechanisms to fill in the other colors. True primary colors, by definition, cannot be made by mixing other colors. So this approach can't be used to create a conventional color wheel, at least one that can perform all of its functions. The question suggests workarounds, which will be discussed below.
The color choice of yellow and purple doesn't really give you two primaries. If you are starting from a RYB base, each of those colors needs to contribute. Purple = all primaries except yellow, so you sort of have just one primary color in addition to black. But that does serve as an anchor point, opening up the wheel to more flexible placement of the other colors.
Blend red and blue with base colors: You can't get to a pure color if you've already blended in something else. You can keep increasing the amount of red or blue so the proportion of the something else is lower, but the something else will always be an impurity. This approach is kind of cheating on an artificial constraint you imposed.
Basically, any color in your model, that needs to be a pure building block color, must be either in the model as a base color, or creatable from true primary colors in the model. You need red and blue in the model. Adding them to something else and then trying to get rid of the something else doesn't serve a purpose. Just put them in the model cleanly.
The important thing is getting the right colors in the right places on the wheel. The idea of starting with a 2-color base (really a single primary), is a mechanism to give you more freedom in assigning colors to the other positions on the wheel. Once the wheel is completed, it isn't all that relevant how you created it; it will be a finished tool. Use any method at your disposal; don't artificially constrain yourself.
Blend external colors to build gradients: This is essentially the same as the previous bullet. You can build gradients this way, but you can never get to the pure color. The model needs to start with the two pure color endpoints and then create the gradient in between.
Trial and error mixing: This sounds like the idea is to work "offline" to create the perfect color, then plug it into the model. Sure, that could be a way to create/select a color. It would be faster to do it mathematically, but your objective is "looks good", and there isn't a math function for that. Trial and error is a logical way to get to "looks good".
Calculate color distances from the wavelengths: This sounds like relating the color distribution to the spectrum, which is in some of the suggestions I'll mention below. That seems like a great place to start. But there's a crazy twist on this one.
Color actually exists only as an internal sensation. The vision system translates light wavelengths to create it. So your perception of color is based on wavelengths, ergo, color equals wavelengths. Oooh, I finally got to use "ergo" in a sentence. The mathematical color models abstract away the wavelengths. So if you want to build your model on top of, or with the aid of, an existing color model, you'll need an additional mechanism to relate it back to wavelengths.
If you don't need to force certain colors into certain geometric positions on the wheel, there are all kinds of things that may improve the harmony of the displayed colors, or at least would be avenues to explore.
You can think of a conventional color wheel as the continuous color spectrum wrapped into a circle, then distorted to forcibly center the primary colors. Then it displays a 12-color resolution of it. Using proportions that more accurately represent the spectrum may look more natural. Wrap the color spectrum in a circle, superimpose the wheel on it, align it as desired, and make each position the color centered there.
Building on the previous bullet, you ought to be able to rotate the wheel so it is centered on different colors. In fact, if you just overlay the wheel on the actual spectrum, I'm guessing that color complements wouldn't be aligned, and you're playing with color spacing anyway. There wouldn't be a special consideration in using the centered colors. That would provide some flexibility in the exact colors to select.
The gist of the objective is to get better distances between the colors. There are a number of ways to do that. For example, find better colors for the positions on the wheel, find better positions for the colors, and/or increase the number of positions (you mentioned in chat that the number of positions needs to be 12, so that option wouldn't apply).
Can a harmonious color wheel be used to produce harmonious colors?
(I'll refer to the color wheel producing colors, but really it's producing colors from base colors of the color wheel.) Any color wheel can be used to produce harmonious colors, just select harmonious colors to produce. Assuming the color wheel is built on a reasonable collection of base colors, it can create all kinds of colors.
Consider that almost the entire spectrum can be represented in the existing color models using three primary colors. All the colors are in there. Some look good together and some don't. The color model and color wheel let you create them all, regardless of how they look together; they're all just colors. Beauty is in the eye of the beholder.
The stated objective is to identify colors that look good together. The reference to a new color system suggests that, rather than simply a single pallet of harmonious colors on the color wheel, the objective is a tool that produces only colors that look good together because of its base colors.
The project is focusing on the tools that create all the colors and trying to make the tool, itself look good. That suggests that if the tool looks good, any colors it creates will look good together. But consider that if you were going to put up some shelves and wanted them to look good, it wouldn't accomplish that objective by painting the screwdriver a nice color.
Colors that are not primary colors can be created multiple ways, mixing different combinations of primary and secondary colors. To produce the target color, it doesn't matter how the true primary colors are combined in the base colors as long as the mix contains the right totals. That's why, for example, you can produce brown from red + green, red + blue + yellow, orange + blue, purple + yellow, etc. As long as the base colors in your model contain the needed true primary colors, the appearance of the color wheel doesn't really make a practical difference (other than the purity with which it can produce the desired colors).
By creating a tool that looks good, you're replacing a model based on true primary colors with a model based on secondary colors. That greatly reduces your ability to create pure desired colors because your base colors are true primary colors that are already adulterated with additional colors.
So this approach only hobbles your ability to create colors. It doesn't help with an objective of producing colors that look good together (unless the idea is to create a good looking pallet and then use that to identify rules that could be applied more broadly).