Some earlier CC cowboy hats and related items were made exclusively with black hairs and assigned to Custom. What I had in mind was getting or making the full range of hair colors (at minimum: black, brown, blond, red). My first approach was via the above tutorial, but I found it very frustrating and it never quite worked (all the steps could be followed, but the resulting hairs never showed up, not even as Custom).

So that left me with exporting the base game Cowboy hat (times four, one for each hair color), copying over the CC hat textures, reimporting and saving, and using Cat's (not Theo's) Hair Binner. Granted it's simpler, but it seems a lot cruder and there is no option for different hair styles (as far as I have been able to figure out, yet).

You first need to recolor the hat-hair, to get the files for all 4 hair colors (make sure it's SavedSims files, not Project files, or it won't work).
You also need the hair you want to associate the hat-hair with (I'd assume one of each color).
Then you can try the tutorial again.

There are other ways to do this, too. You can for instance link up the other hair mesh (extract from ingame without doing any changes to it) via the 3DIR and then change the textures and all that. It does often involve a bit more fine-tuning (you need to know how to make new hair files, link hairs, and know how the resources work, because there can be naming issues).

Personally, I prefer having the hairs and/or hats available for all categories, and setting them up ingame (Seasons+), because then any hairstyle can be chosen. Much less troublesome...

I guess what I really want to know is, how to turn the hat into an accessory so that it can be used with any compatible hairstyle (obviously, some won't be, but some will). One person seems to have figured this out, but I'm not sure how. Nobody else has, they just recolored the hats and pretty much let it go at that.

There's really no difference in making a hat accessory than any other accessory. Extract the mesh and texture from the hair files, and use an accessory tutorial. You may want to make the hat a bit bigger and slightly higher on the head so it's compatible with more hairstyles, but that's it, really.

Accessories have a lot in common with making new hair/clothing meshes, so if you've made any of those, you're halfway there already.

(Tip - there's a workshop starting next week - May 18th, 2024 - centered on accessories, where you can join in and ask questions. If you miss it, they post videos here afterward)

Will do some research on this and hope for the best. (Used to work with Milkshape back in my Sims1 days, but my copy is outdated now, so I hope I won't have to do any mesh work.)

Sorry, meshing is required for making accessories (you may be able to use Blender, but there are fewer tutorials for it).

I'd recommend using Milkshape 184 or 185 (I had some troubles getting 184 to work, but 185 has worked fine so far).

My understanding is that you can't legally register Milkshape any longer because no one is minding the store, nor has been for a minimum of 10 years. This is problematical if you wish to use it for longer than 30 days.