The lever will move back and forth, but it just sort of flops to one end or the other. There is no friction at all. It clearly needs to be attached to something, but I don't have another one of these handy to compare. I've searched to find a photo, but that isn't an area people tend to feature when they are showing off their collection.

Hmmm, so in the following photo, are you referring to the red or blue arrow?

Ahh yes !  This is a well-known Pre-war Remington disease unfortunately !  In other words, a built-in design fault.  There is a tapered stud that is rivetted into the linespace lever and rides in that arcuate slot (blue arrow).  On the inside of the slot is the linespace pawl and return spring.  The stud has snapped clean off so there is now no connection between the linespace lever and the linespace pawl, therefore no linespacing and the lever is flopping about.  There are two solutions.  Obtain a parts machine and transfer the linespace lever complete. Or, remove the linespace lever and take it to a precision engineer to have the remains of the stud drilled out.  I had this done for a customer's machine, and he had to use a cobalt drill to drill through the hardened stud.  It is most important that this job is done accurately because if it is out even slightly, the new stud will not ride in the arcuate slot.  Having removed the stud, tap the hole - I think we used M2.5 or M3.  I was lucky enough to have a metric stud that came off an old Adler golfball typewriter that I could screw into the tapped hole in the linespace lever.  You might have to get a stud specially made.  Virtually all Pre-War Remington portables, and a few post war ones, use this horrible brittle stud, which breaks frequently.  I've seen quite a few !

Good job! Post a photo of your repair if the mood strikes you; it will serve as inspiration for others, and prove that a little ingenuity can often save a machine.