Ok, this will be my last attempt to clear some things up. Just forget everything I've said so far and see it from this perspective. I know you're not interested in using it, and don't want to change (understandably so, after 23 years) but this is just to give you a different perspective. This reply isn't about me telling you how great my system actually is, it's more like a conversation on the note naming systems and their functions.
Among others, music theory has two functions: To help people play and orient themselves within music, and to make it possible for people to communicate concepts. I'll name these two things the cognitive side and communicative side. It is not necessary for these two sides to have any relationship whatsoever. You might be visualizing dots in your head while playing a melody, but if you had to communicate it to someone you could also have the music theory knowledge to tell them the key and scale degrees of the melody. I could provide more examples, but you probably get the point - you don't always think in terms of the theory you use to communicate while you are playing something.
My main argument is this: I think the letter naming system is good on the communicative side, but bad on the cognitive side. For example, say you're playing in Ab major and you had to think of all the notes in terms of their names while playing a melody or chords. You'd have to know all the notes within the chords of that scale so well that your thinking is almost on a subconscious level. For one key, Ab major, that will take a lot of practice, so what about the other 29 major and minor keys? Now I know, no one actually thinks of every note name while playing and that's precisely because of this problem. It's too overwhelming and inefficient, but when used a communicative device, the letter naming system works great because you can talk about absolute note names regardless of the key you're using. It's specific. This whole paragraph can be summed up by this
To clarify, I'm not implying people actually use the letter naming system as a cognitive device, I'm explaining why it would be a bad one to use. Also, in the video Neely says that we should ideally think about nothing when playing. I agree, but sometimes we have to think about something (like he mentioned) and I would rather think about something less complicated than something more complicated.
So what's the alternative? Assuming you're working with the constraint that you're only playing in major or minor keys, assign numbers to scale degrees. 1234567. No matter which key you're in, you only need to know one formula for each chord. 2461 will always be the ii chord. When thinking in letters, you have fifteen different formulas for the ii chord in a major scale. When thinking in numbers, you have one. That's why it's more powerful as a cognitive device, it's because it requires less thinking. This way of naming is already a well established system. Everyone knows it and it's a very intuitive way of naming notes.
Note these two systems - letter naming and number naming - can both be used because they serve different functions to musicians. Even though there is some overlap in function, one is used better as a cognitive device and the other better as a communicative device. So far everything I've said seems fairly obvious and uncontroversial. Where does my original post come in then?
When you start playing with notes that aren't one of the seven numbers, you have to use #'s and b's to specify them. This leads to things like b2 b4 b6 b1 for the bii chord (if you wanted to keep the 2461 formulation consistent). This bothered me because b4 is actually the note 3 and b1 is actually the note 7. Because you can have different names for the same notes, it seemed like a slower method to use as a cognitive device. That is what I meant by "inconsistencies". Therefore, I thought it would be better to give each "b" or "#" note it's own category, using a letter to name it (1 i 2 n 3 4 t 5 s 6 r 7). People raised two objections to this:
It makes communication harder because people aren't familiar with my naming convention. Response: I don't use this system to communicate, I use it as a cognitive device, a way to think of music. This goes back to the statement that communicative function and cognitive function do not have to share a system.
Getting rid of enhamonic names isn't a good idea because different names can refer to different note function e.g. #4 isn't always the same as a b5. Response: I agree with this, but I honestly just don't care enough about it. It's such a small thing to worry about. Even if I did care, I don't see how it's relevant since my system isn't primarily used to communicate compositions.
Saying my system is a simpler cognitive device is subjective of course, but I'm still making an argument trying to prove it might be simpler.
So just to recap my line of reasoning: 1. Music theory has two functions: Cognitive and Communicative 2. Certain systems are better suited for different functions 3. The system I proposed was only meant to be used a cognitive device, not a replacement of the letter naming system. 4. I guess it doesn't really matter whether or not you want to keep enharmonic names, for me it's just easier to discard them.