There are 2 main aspects of this effect.
The first, as you noticed, is the keep the trail’s texture from “moving”. As that thread you linked to discusses, this is harder than it sounds and Unity’s trail renderer actively works against you in achieving it. The only real solution is to not use Unity’s trail renderer at all. There aren’t even any assets on the store that I’m aware of that are able to do this properly. I’ve ended up hand writing my own every time I need this, even excellent assets like Ara Trails doesn’t have this option. This is obviously a big issue, but it is what it is.
The second part is the trail mesh is a constant width, and the texture being used is a gradient height map that doesn’t tapper. Instead the trail is fading out the alpha and that alpha is driving a cutoff edge on the height map. It’s a carefully controlled dissolve shader. Something like this: