I finally managed to find good information about getting rust off of a cast iron woodstove by using 
Marginalia Search Engine, a specialty search engine that is intended to resurface the "old web" of private websites and bulletin boards and stuff instead of SEO and corporate slop. 
A few years ago in the winter when we were using the cast iron woodstove sometimes, someone (me) uhmmmmm absent-mindedly left some candle holders sitting on top of it with candles in them and those included ones carved out of solid blocks of pink rock salt (hideous, they belonged to my MIL, who was addicted to candles. Why didn't we just get rid of them? We hated them. Natural aversion to throwing things away. We have since thrown them out). So it turns out that ummm the candles completely liquefy if you do that and then light a fire in the stove, and they like cause the salt to run and melt onto the surface of the wood stove and salt is bad for cast iron. So. Big rust spots. 
And the rust spots have got worse with time, because when it first happened and we tried to get them off, we tried with normal google and duckduckgo searches and got no better advice than sandpaper and steel wool. We only managed to get a tiny bit of the rust off and determined that getting it all off would have taken about 5000 hours of hand-sanding. Since that was not a worthwhile proposition, we left it that way for another year.
So anyway, I tried Marginalia a month ago or something, and it only took a few minutes to unearth a 
thread about restoring cast iron woodstoves on an old-fashioned bulletin board on "finishing.com, the home of the finishing industry". It's straight out of the internet 20 years ago. And the information was MUCH better!
- WD-40 softens rust
 
- wire brushes, not sandpaper or sandblasting (although industrial, like, having the stove ripped out and taking it to someone who will sandblast it is the nuclear option if it's completely covered in rust everywhere)
 
- wire brush attachments for power drills
 
That was all the info we needed! WD-40 never seemed stinky to me when I was using it on door hinges and stuff, but when you spray it over the visible rust on a wood stove it is noticeable, though not TERRIBLE; it smells kinda like you're in an auto shop, but not in the middle of the car part. Like by the entrance.
You can get visible change on small rust spots with a handheld wire brush. A few hours on two days with the drill attachment has seemed to do the majority of it. It's very hard to work in eye protection goggles and a high filtration mask though. I have to stop, lift the glasses to look, then lower them and start again every minute or so. We are not planning to repaint the spots that have been taken back to the silvery iron, according again to the advice on this bulletin board. Apparently lighting a fire after the WD-40 is already going to be stinky enough and the paint would be worse. You can get protective stove polishes of some kind apparently. 
This stove is a Jøtul 3 Classic cast iron woodstove, in a traditional 19th century style. It's completely inappropriate for this 
1950 modern-style house. The expected stove in the livingroom is (and no doubt was) a masonry stove, which is much better at heating an area because the ceramic conserves heat and releases it gradually. The form of masonry stoves, which are of course built on-site, was typically streamlined in the years after this house was built. Nowadays you can't build them yourself anymore and that makes them more expensive, so somebody probably replaced the original one when it failed with this cast iron stove perhaps in the 1980s, which was the last time this model was made. But crucially, although a woodstove is completely inappropriate to the house and less functional, there 
were and are woodstoves that are more minimal and modern in form and they could've just got one of those. But nope. 
Anyway, we can't afford a masonry stove like, ever, but our ambition is to replace this woodstove with a 
Porin Matti, a cheaper alternative to a masonry stove that is still slightly better at retaining heat than a cast iron stove, and which also (a) was in popular use in 1950 and (b) looks similar to the style of masonry stoves typically found in our type of house. These only cost about 2500€ (not counting labor), in contrast to masonry stoves which are typically over 8000€ not counting labor (and requiring much more labor because the mason has to build it on site out of blocks and tiles). We would've been able to buy one this year probably if we hadn't had this broken sewage pipe issue, which ended up costing around 10k. (We had previously earmarked that money, an inheritance from my great-uncle who died recently, for restoring the outer front door and maybe a stove; but the last of it got used on the plumbing instead.)