I'm not familiar with the NES in particular, but typically the software is synced with the video signal via hardware interrupts.
I don't think it's possible to create a mechanical way of moving the gun fast enough to achieve the needs of even a low-res TV screen. If it was a physical mechanism, you can imagine it'd break all the time.
(Progressive versus interlaced is the difference between the "p" and the "i" in 480p versus 480i, 1080p versus 1080i, and so on.)
Therefore, with interlaced video, each screen update, called a "field", only has half of the data of a complete frame, and the other half is not sent: The next field is one-half of the next frame. Therefore, interlacing cannot be undone cleanly. Deinterlacing, and it's called, is inevitably a process of reconstructing an image which has passed through a rather brutally lossy compression algorithm.
(Distributing interlaced video where the same frame is broken up into two fields, and therefore frames can be reconstructed cleanly, is called "progressive segmented frame". It was used in film-to-video transfer.)
CRT TVs were designed around interlaced broadcast signals, but CRT monitors most certainly supported high resolution progressive scan video. The much vaunted Sony GDM-FW900, for instance, one of the last great CRTs before LCD monitors took over, supported progressive scan resolutions of 1920x1200 at 85hz.
And the NES and most early game consoles really did output a progressive scan signal that CRT TVs could display, through a hack known as "240p". Basically, the device only ever outputs odd fields in an out-of-spec NTSC signal, giving the effect of a progressive scan image with half of the vertical resolution. CRT TVs, being "dumb" analog devices, would just keep redrawing over the odd lines every time, and never ever touch the even lines.
This is the source of the "scanlines" effect of darkened horizontal lines spanning the screen, as these lines were never directly hit by the beam while a 240p game was running. It's also part of why these old games look like crap on modern LCD TVs, as almost all of them have scaling hardware that is ignorant of the 240p hack and incorrectly treats the signal as if it were interlaced, producing tremendously ugly results.
The NES, like virtually all home computing hardware of the 70s and 80s, operates at 60 hz progressive scan of 200 lines, give or take a few. That is, of course, the same bandwidth of the 525 interlaced lines of broadcast video.
In those days, the horizontal and vertical resolution was bound by RAM capacity rather than pure video signal bandwidth.
In either case, the screen is updated at 60 hz.
Hook up an NES to an old CRT and look closely at the screen: The scanlines are absolutely still. (This is also the case with the full-screen menus of VCRs, with the white text on blue background.)
The NES most definitely outputs progressive-scan video at 240p, with all 240 lines drawn 60 times per second.
