To be fair excess fats are bad for the normal drain too. Temperature is a good point, my initial idea was that the channels would be along the grass paths so if a bit of grass dies by the opening it isn’t really a massive concern. By the time it soaks in any real distance it should be cooled by the surrounding soil.

Bath/shower usually is the cleanest source to go for first, the pipes are next to each other for me. Either way I would set it up so that they can be easily removed when I don’t need the water going onto the garden.

Do need to check how flat the area is, exit point from my house is pretty close to the ground so a slight elevation change could be a problem.