Found out that our new, faster router can have three networks = two 'primary ones', one that puts out a stronger signal but in a limited area, and one that puts out a weaker signal but more generally accessible, and a guest one which can do the whole internet connectivity thing but won't give access to local files. Fine.
Turns out that printers can only connect to one network. Your device, like your computer or tablet or whatever, has to be on the same network as the printer; otherwise, no go. The printer might as well not exist. According to a brief search I did, lots of people think wait, WHAT? Only one network can have the printer, unless you want to do some magic or have a dedicated print server or whatever?