So I've been messing around for the past week or so with creating chord progressions using techniques that fall under something I'm calling "mechanical harmonization" (appears I've coined the phrase!!?).
 
A couple of my methods involve using an arranged alphabet - either the standard alphabet, or for example a letter frequency chart that lists letters from most used in the English language to least used.
Then I'll take each letter and apply a note name or chord to it - A = Amaj, B = Bmaj and so on, for example.
I'll then go ahead and plug in a short phrase, like "Hot Pizza" and convert it over to chord names and viola, a new chord progression has emerged!
 
Another thing I'm trying is using a virtual 7-sided die; I set up a chart of 7 note names (for each note in the C-major scale from A-G), followed by a list of chord qualities (maj7, min7, 7sus4 etc etc., and then perhaps a 3rd list for some other arbitrary thing like, if it lands on 2 or 6, the chord is raised by a half step.). This yields... interesting.... results. Depending on the choice of chord qualities, the sound is more or less too jazzy, atonal and non-functional for what I want.
 
Nonetheless, I'm finding it tough - almost impossible - to find something useable with these "mechanical" methods. I'm interested in finding something that automatically generates the patterns that my ear would be drawn to, but at the same time isn't something I'd be inclined to write in the first place... but that also isn't completely random!!
 
Anyone got any good mechanical harmonization games? It's really interesting and I'm open to different techniques that may make it possible to create something more interesting.

That's probably because there are a finite number of chord progressions that will sound good in any given genre.  Probably easier to learn what they are and then make the most of them.
 
Popular chord progressions and the songs that use them:
 
https://www.hooktheory.co...mon-chord-progressions

+1 to the above. The finite aspect of this being useful is also compounded with song structure, tension/release, etc. Ironically, one of the bigger gluing components is the vocal, both for tonal character as well as the words. Chord progressions alone tend to fall into defined sets, the melody/harmonies embellish them.

I had thought to do similar naming the kids with constants/vowel randomness, but Zeukerab, Kilktuoce, and Hufram all got shot down! (kidding, don't tell Hufram!)

It depends on how interesting you want your chord progressions, and also what style of music you're playing.
 
So assuming you're not doing jazz, you could try:
 
1. Take a fairly standard chord progression, then change chord IV to II, and V to III. So for example, if you're playing in C major, change F to Dm or G to Em. Don't do it all the time though, swap and change.
 
2. Take a one note from the first chord, then try another chord with one note that matches. So you could take G from a C major chord, then you could go to Em, G, Gm, E major, Ab Maj 7, Bb 6 etc. Then do the same for the next chord.
You can also try this using notes from the melody.
 
3. What I call "circle of 3rds" (as an alternative to circle of 5ths). This is good for prog rock... play Major chords up in Minor 3rds. So C to Eb to Gb. Or do Minor chords in major 3rds - Cm to Em to Gm. Mix these about. You can also could try dropping down, or going up a semitone (but in the same key as the current chord). So if you're on F, drop to Em, then try going up / down a 3rd on the next chord.
 
4. All of the above, but also change the inversion of the chords (e.g. C with an E bass, or G bass etc).
 
Another suggestion is using Liquid Notes: https://www.re-compose.com/liquid-notes-music-software.html. This re-harmonises chords, melodies etc.
 

Musical composition using (semi) random processes has been around for a long time - https://en.wikipedia.org/...ikalisches_Würfelspiel

One way I find useful to come up with ideas is to use a hardware sequencer. Set it to play a chosen scale, set the pitch controlling knobs without paying much attention to where they end up, at least at the start, then use the step on/off, jump, octave, reset, direction of playback etc. switches and see what emerges. Then tweak the results as seems appropriate.

In software Sugar Bytes Thesys can do much of the same sort of thing, as can Cake’s more limited step sequencer.

There are also quite a few plugins that can generate random MIDI notes from a scale and key of choice or randomise existing MIDI.

A quick PS-

I’ve not thought this through, but as far as relating notes to the most commonly used letters (or words) goes, Zipf’s Law might have some interesting implications.

https://en.wikipedia.org/wiki/Zipf%27s_law

Have you read any Howard Gardner Frames of the mind, or Douglas Hofstadter Metamagical Themas "Questing for the essence of mind and pattern?  These books sound like they might be right up your alley.  I use randomness for Biab Musical styles and than base my chords randomly than just try to record on notes that only exist in the rhythm.  I don't add notes outside of the rhythm I'm using.  But outside of Biab it takes a lot more work just to push the drum parts across to other instruments and in a different octave.  and from there I peel away the different parts keeping only 1 note per instrument.  It's very time consuming to compose this way, but it ends up being very different.