Okay, this will take a bit of work to get it to be a quick easy 5 minute job in the future.

Mix up baking soda and water in a old dishsoap bottle or whatever you can find that you can squirt or sqeeze the mix onto the grout . Think the thickness of "gravy" not too runnny you want it to stay put .. Squeeze a layer of this on all the grout , I say let it sit a few hours or over night..

Because I am a freaking idiot OCD about cleaning as you know

I would take a dish rag and a pan of hot tap water and by hand just wipe the mixture off, yes soundls like a lot but this is a one time deal a year usaully... just keep changing the water out until its gone. Then spray a light amount of a degreasing spray on the entire floor and mop.. again use hot water you want the floor to dry quickly.

Your grout will be clean clean clean and be bright, your floor will sparkle. Then once a week just sweep the floor well and run a hot tap water damp mop over it and its done.. Of course wipe up any daily spills when they happen and your weekly floor cleaning will take just a few minutes.

Baking soda is cheap and a bottle of degreaser I prefer the cheap Walmart brand. Both are cheap , the degreaser will last you a year