Hmm...looks like some of it does need to be refinished. Any place that is no longer shiny needs it. I guess I would explain this (gently) to the owner, and explain that a "deep cleaning" cannot be done on areas that are scuffed and non-shiny now. They CAN be cleaned, but surface only.
There is a lot of disagreement on how to clean wood floors. Some people use steam mops. I would never do this, because warm steam can warp and damage wood floors. I would first vacuum thoroughly. I only use vacuums with really bright headlights, which shows me where there are spills, pis and damage, etc. If food has been spilled, it must be removed before attempting to mop a wood floor. I use only a gentle all purpose cleaner to do this, using a damp clean sponge. Once this is done, I would mop the floor, using a damp mop - I use Swiffer mops but not their icky pads. I use ordinary white washcloths which fasten onto the mop easily. For wood I use a cleaner made specifically for wood floors. I dribble a bit on the cloth and start mopping, with the grain of the wood. When the washcloth stays white, the floor is clean. The problem with an older, scuffed floor like this is that icky stuff is going to be stuck on and especially between the wood planks. Sometimes I use a pocket knife ( a dull one) to scrape this stuff up. Or a long handled small screwdriver...anything that will not damage the wood.
Wood floors are labor intensive, and there is NO way around that. Good luck with this project!