No, you're right. Old-school radar displays were basically like oscilloscopes where the X and Y position were controlled by the angle of the radar, and the current range of the radar return. So it's a type of polar plot.
The phosphor afterglow made it so that stronger radar returns remained on the screen for a bit. If the radar made more than a revolution in that time, you'd see the same airplane as a new dot ("plot") that had moved a bit. You could use a felt tip pen to mark the plots as "tracks" on the screen.
There were also special radar screens with a movie camera pointed at them, where hours of radar returns could be recorded for later playback.
For instance this sped up recording of Warsaw Pact planes during the 1968 revolution in Czechoslovakia: https://youtu.be/rAUodXI4LPw?t=622
They were vector displays. But possibly not quite in the way you're thinking.
The original radar displays would scan from the center outward along a radial. The timing of the scan was predefined to scale for distance. The beam intensity signal was directly the (amplified) radar return signal. So a stronger returned signal would cause a more visible "blip" on the long-phosphor display.
The interesting part is to make the radar beam scan around the CRT display, the whole cathode tube emitter assembly would be driven by a motor synchronized with the spinning radar dish. This rotation would have to match the speed and direction of the radar dish at all times, otherwise the blips would show in the wrong place.
The fixed radial and distance lines would be printed either on the CRT tube itself or on a transparent cover. Displays like this were used for decades, probably well into the 1980s or even early 1990s. Newer versions were able to use simple electronics to scan in the X and Y direction independently, to avoid the more complex rotating beam emitter assembly.
I was wondering about that after asking my question, since converting the signal to an x/y scope requires trigonometry. Probably a challenge to do electronically in the early days of radar. Leave it to the WW2-era engineers to find an electromechanical solution!
Radar was surprisingly advanced by the end of WW2, the Brits and Americans had ground scanning radar by the end of WW2 (H2S and the later H2X) all done without transistors or computers as we'd recognise them :).
Just to expand on one point the other two replies touched on: The beam does indeed scan radially outward from the center (though the angle can either be produced by physically rotating the deflection plates, or computing sin/cos electronically and applying to the x/y plates), and the amplitude of the returned signal does directly drive the beam current (so a target with a larger return will appear brighter.
Additionally, there's also a storage tube effect going on, too--but it works like the variable-persistence mode found in certain oscilloscopes, not the bi-stable mode found in storage-tube x-y vector displays for early computers (some o'scopes had both var persist and bi-stable modes).
One consequence of the above is related to early attempts at designing stealth airplanes, like the SR-71 and XB-70: The beam current is set such that the radar tube tends to build up the "blip" over multiple sweeps, and the blip usually moves less than its own diameter between sweeps.
But, if the airplane has an intrinsically low radar cross-section and if it is moving so fast that the blimp moves a greater distance between sweeps... the radar operator may just interpret the little ghost blips as noise, especially if he or she is overwhelmed by a large number of targets.
Small detail question: Did CRT radar scopes really have scan lines? I would have guessed they were vector displays.
Re. list of extra features: Since the app targets planespotting, it would be cool to show aircraft type, maybe for a few seconds after you tap a blip.