I highly suggest you check out this book: https://www.amazon.com/Darkroom-Cookbook-Alternative-Process-Photography/dp/1138959189/ref=pd_lpo_sbs_14_t_0?_encoding=UTF8&psc=1&refRID=7X28P62RGESTJJRKT4MD

The history of film development is a long, experimental one. Even today, people continue to experiment, even using coffee to develop the film (https://en.wikipedia.org/wiki/Caffenol).

The point of each chemical is the step it is made for. Developers develop film. Stop bath halts development. Fixer removes all developing residues and undeveloped silver and "fixes" the image in place. Permawash removes the fix. Washing for a long while removes both of those last two. Wetting agents help to get clean drying.

The point is, there's nothing about any of these chemical steps that is manufacturer dependent. The chemicals themselves are agnostic - they don't care who makes them.

So, you are perfectly in the clear to use a developer from Kodak, home-made stop bath, fixer from Ilford, perm-wash from Heico, and wetting agent from Kodak again. No harm will befall you from jealous chemicals, because chemicals cannot be jealous.