Posting my experience in case of value to others. In the end I did the whole thing with a random orbit sander (Erbauer, 400W power, 125mm disc), using only 180 grit discs. This worked very well for my situation.
In much more detail…
They're pine floorboards in a Victorian terrace, almost certainly the original boards. They were very dirty (grey ingrained dust) and covered in paint splatter. Lots of minor imperfections (light pitting, cracks, knots and missing knots, etc. etc.). But surface actually very smooth under the 100 years of grime.
I bought two sanders, from B&Q: the 400W Erbauer random orbital, and a more powerful 900W Erbauer belt sander. In fact the belt sander wasn't much use: where it made good contact with the surface, fine, but all the boards have slight ridges and slight troughs/depressions, and it didn't get into the troughs. By contrast the random orbital is able to sand much more adaptively.
I started with the standard approach (80 grit aiming to progress up). But then I realised that 180 grit was taking away as much as necessary, almost as rapidly. No need to go through three grits, and with 180 the surface is already very smooth. And what-you-see-is-what-you-get: you're immediately seeing the final surface, if there's an imperfection you can accept it, or keep sanding until you're happy. The sander gets in very close to the edges. I got the discs from Screwfix, cheaper and better than B&Q I think. You can get 10-packs, B&Q only do more expensive 3-packs.
This is an unorthodox approach, I know, but it worked for me. It was a bloody horrible job (dusty, noisy, very time-consuming = 3 days of hard work)... but I think that's probably true however you do it.
I note my previous experience with a hired floor sander, on a 1950s oak parquet. I found that the sander gouged marks in the wood; it was awkward (like mowing a very small lawn with a large lawnmower); it didn't deal effectively with troughs and depressions in the surface, leaving them unfinished; and because it was hired I felt under pressure to complete the job quickly. Similarly: in my present house, another room was sanded by the previous owner or a tradesperson (presumably by the standard technique): a) there are multiple gouge marks, and b) trough areas in the boards are not properly sanded. My results are frankly much better.
I note that a particular problem in my case was very old carpet adhesive strips, impossible to scrape off, needed to be sanded off, and this then smears the degraded adhesive, which in turn needs to be sanded away, gunging up the sanding pad.
So for a small room with old floorboards, I'm much happier with 400W random orbital using just 180 grit (than with standard approach, i.e. floor sander or smaller belt sander starting at 80 grit). If I did it again, I suspect I'd use 120 grit instead, to get just a little bit more bite. You need to be on your knees and sanding with close attention to detail, often rocking the disc on its side to get into hard-to-reach areas and deal with more recalcitrant material. You need to change the disc often. As pointed out by the person who advised me, very important to remove or hammer in nails, screws, staples. And to wear a mask and shut doors: even so, the whole house gets covered in dust.
Finally, I'm no expert, so if anyone disagrees with my approach please feel free to say so!
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