Perhaps Scottish grout is more permeable. I have never had a problem with the grout staining (or me staining for that matter.). You get staining when the dye stays in contact with something, be it countertop, hands, grout, whatever. The shower water washes it off before it can stain. I wear gloves because my hand stays in contact with my hair and the dye as I am painting it on. However, if a bit runs down and I wipe it off immediately - no stain. If I wait, I need to use isopropyl alcohol to remove the stain on my forehead or neck or wherever.
Can't say I've ever used a jug to rinse my hair. Sounds like a lot of refilling would be needed. When I use the sink to rinse, I use the sprayer. Because my head is tipped forward this increases the chance it will get in my eyes. In the shower I always tip my head back so I won't get dye or shampoo or conditioner in my eyes.
As I said earlier - I've been grey since my twenties. That's 30 years of hair dye experience. Grout, skin, eyes and hair are all fine.