and seemingly nobody else has done this before where I could copy the approach.
It's not that it's not doable, it's just that depending on the data used there are many nuances, and that usually other approaches are better.
For example, extracting clean vectors out of possibly highly erratic rasters is an extremely difficult task. You can use many 9 facilities to assist in that task (see, for example, the Trace template in the Transform pane), but reducing it to a "copy this" approach is not realistic.
That's not just a 9 deal, by the way, as many books and papers have been written on the ins and outs of extracting vectors from raster images. The usual problem is that you end up with so many errors that it would have been easier to vectorize manually than to hunt for and replace errors that are in the result of an automated process.
An example of that, tons of errors despite having unlimited budget and using the latest techniques, is in the user manual here. That's one reason there are so many contractors in low cost labor areas around the world who specialize in manual digitizing.
Instead of trying to extract vectors from raster data, it is usually much better to invest time into finding the vector data set that was originally used to create, say, the WMS.
That might not always be possible, because the reason some organizations (not all, but some) publish vector data in raster form via WMS is precisely because they do not want you to have the vector data. They want to keep it for themselves. So they publish it in a form which deliberately obfuscates the data, so you can see a view of it that they prepare, but you can't get at the actual data itself.
If despite their disinclination to provide you with the data you want to grab that data anyway, by scraping the WMS tiles and then reverse-engineering from raster to vector to try to break their obfuscation, you have your work cut out for you.