I've been researching. Lots. Agreed, linseed oil isn't the worst finish to have. And the flooring that isn't stained is beautiful.
@nereo Sanding isn't a good idea - the wood is pretty thin already. If I can improve the staining sufficiently, that would be better for me. Let a future owner worry about having to pull out the floors if they sand too far! As for why it had carpet, I'm not complaining too much. Two other rooms the floors are basically in perfect condition, if dirty (they didn't have the 60s carpet). I just wish they'd pulled this carpet 20 years ago.
I did a bunch of scraping up of the dirt earlier, there's a bit more but I got most of it, and swept it all up. That helped quite a bit actually. I think my plan of attack next is to scrub with dawn. The floors are filthy after being under carpet for decades, they need a good scrub. Once they're clean I will be able to see more of what I'm dealing with. (This is going to take a week or more - lots of area to clean.) If I'm lucky, the dishsoap will help with the staining, but even if it doesn't getting the dirt off is good. Right now the floors are gritty and I do not want to sit on them.
From there, I will consider TSP and spot cleaning.
@ysette9 , good call there, it appears that TSP is relatively safe for wood. Given the care needed to use TSP, I don't want to use it on everything, just the stained areas. I may also see if a layer of baking soda rubbed in would help lift the stains too, I've seen that mentioned a few times. But I'll determine the exact plan once I get the floor clean.
I do think there's at least one area with urine damage. This room will have an area rug so I can hide a variety of problems. But this staining is too extensive to hide.
Some pics here (the scale is weird, I have no idea what I did). This is before I pulled the staples and did a bunch of scraping up of dirt, so it looks a little better now.
https://imgur.com/a/w0zjrp6