So, businesses that post videos of themselves cleaning floors like this seem to use a commercial, truck-mounted hot water extractor. These are ... much more powerful than anything easily available to a consumer, because they use strong detergents, near-boiling water, and very strong suction to basically wash and dry the floor in one pass.

If I was in this situation, I'd shop around a few cleaning companies that have a truck, and then see if they'd give you a discount for coming out 3-4 times a year on a schedule.... or I would look into sealing the tile with something clear but self-leveling so that you can take care of it yourself.

At this point, I'd try some of that keyboard cleaner goo ... it would get in the crevices and then lift the dirt. It would be super time consuming though...

You can rent a circular saw or a table saw from the hardware shop for pretty cheap ... and if you don't want to, a basic saw and miter box are pretty inexpensive as well.

Otherwise, triangle molding would work to a degree, like another comment said.