I really do like the Idea of direct painting on a 3D object though.
But is there something that does that and is not a whole 3D package
like 3D coat, which looks really good but too expensive and extensive.
Well, wasn't the idea of this exercise to do it the "hard way"? Can I let you so easily from the hook?
But, yes, there are such apps, probably a ton of it...
The most used is Substance Alchemist (formerly substance painter), that let's you create prb-materials ... and paint on a mesh. It comes at a price, though, and newly is subscription only.
The free alternative (with selling their scans of objects in mind) is Quixel Mixer. You don't need to buy their stuff and can paint on a mesh or create semi-complex materials (it's still far away from Substance, but to reach more or less that quality is allegedly their target). I have tried it out, but never painted on a mesh there (for me, it left something to be desired, not many procedural possibilities). But certainly worth to look at and I intend to use it in the near future for real in some projects.
The 3rd one you already have installed: Photoshop. Funny thing is, while I usually create 2d-maps with it, I never painted directly on a mesh there except for a test many, many years ago (more than 10, if I remember correctly). You can load a mesh with an uv map, alter the map, for example reduce it (which results in more distortion). I can't remember ever reading somewhere in a 3d forum that someone mentions the use of it, it's seldom even mentioned that it has that ability, but i genuinely don't know if that comes from a lack of features or if it's just not that much known (you can even render your objects there. Back when this was adopted (roughly 15 years ago) it had a rather good render engine, allegedly coming from Lightwave). Again, I intend to try that out with a better computer.
Funny thing is, I never really adopted the "paint-directly-on-meshes" technique, because I don't hand paint textures anyway. It very seldom looks photoreal (for that I'm simply not good enough). I only do all the marks of wear and tear like scratches, stains, texts, numbers, some small details etc. which are for me much easier to do in photoshop and illustrator, always combined with some (nowadays pbr-) photo and / or procedural textures. What I sometimes did, to make it easier, is painting on a mesh to mark what's what, just so I know where to create the intended things in PS. That you can do in Cheetah.
A few times I mesh-painted in Modo, before that in zbrush (but I never really liked the process there).