the composite to HDMI converter has a single switch from 720p to 1080p

Composite is 480i/60*. That is, 60 times per second a blanking interval occurs, then 240 lines of picture are drawn - either the top (odd) or bottom (even) field. This is neccessary for CRT TVs because a 30Hz refresh rate would cause seizures but drawing all 480 lines 60 times per second would be wasteful. Look it up online for details: if you want videos, I recommend the Television playlist by Technology Connections on YouTube, especially the first video.

*Technically, the vertical frequency for NTSC is 59.94 Hz (precisely 60000/1001) to avoid interference between color and audio while keeping compatibility with B/W sets. In practice, you should check that the video output is actually at this frequency; if it’s 60 then every 1000th frame will be duplicated - no big deal usually unless this also swaps odd&even fields. No such problem exists for PAL, which was always exactly 50 Hz.

If the converter only outputs 720p or 1080p (presumably at 60 Hz), all 720/1080 lines are drawn 60 times per second, which means lines are added with some scaling technique, after some kind of deinterlacing happens.

Deinterlacing is basically a task similar to scaling but with key differences:

• You only need to extend one dimension and exactly 2×
• You know what the missing lines looked like 1/60 of a second ago (and 1/60 of a second later), which can be used verbatim or intrapolated from.

There are various deinterlacing techniques that could be used here:

• doubling each of the lines for 240p60 video: easiest and best for temporal accuracy but loses half the vertical resolution; good for NES/SNES capture because it’s already 240p60 (using timing tricks to make the CRT draw lines of both fields in identical spots) but most capture devices misinterpret that as 480i60, arbitrarily deciding which of the 240-line frames will be the top field
• holding the previous two fields for to the next frame for 480p60 video where only every other line changes per frame: good for viewing (this is how playback usually works on progressive-scan monitors) but wasteful for storage, as each field is basically stored twice and compression does not help. Pausing will show two consecutive fields (it will be random whether top or bottom first, depending on the exact frame you pause on) so whatever moved in between will have jagged edges but that’s expected.
• most common, and likely what your hardware converters do: doing the above but only storing every fully changed frame for 480p30 video: this is terrible for video shot on video cameras because the time difference between fields is not accounted for, and you will see jagged edges for moving objects, or even moiré patterns when scaling. However, this is the best option for PAL film scans because they were shot with 24fps global shutter (and played 4% faster at 25 fps for video purposes) and then scanned interlaced so the top and bottom fields correspond to the same moment in time (as long as the fields are not swapped, which is why capture software usually has that option). For NTSC film scans, 3:2 pulldown is used (each odd frame gets scanned for 3 fields, each even frame for 2 fields to convert 23.976 frames/s to 59.940 fields/s) so this technique needs to be modified to skip the correct one from every 5 fields (or average it with the identical one from 2 fields ago for noise reduction).
• computation-heavy: guessing what the missing lines of each field would be using some kind of algorithm for 480p60 video: this is a kind of upscaling technique and as such can produce artifacts but they will be better than what you’d get with the above methods. This will probably yield the best results if you NEED the output to be progressive video with limited bitrate, such as uploading to YouTube. This is the best technique if you want to do some AI upscaling (will not add detail but help with sharpness; I recommend it for YouTube uploads because the resulting HD files will be stored at better bitrate - and especially for PAL because 576p is not available on YouTube so uploading at higher resolution is required to get all the scanlines, and YT’s scaling of any kind of interlaced or badly deinterlaced footage introduces moirè.
• the best: NO deinterlacing! Store the video in an interlaced format because that’s what it is, even though software support is not as good (the software support has always been good with players, and video editors have gotten better but still the capture devices/software often insist that interlaced video could be unsupported and avoid it). You can encode it to MPEG-2 AVIs (terrible option for storage though, bad quality/bitrate ratio) and burn them to a DVD (or use a DVD player’s USB port if present) and connect the DVD player’s composite output (or better, component/SCART at 480i) with a CRT TV for the intended way of viewing interlaced video. Nobody knew about LCDs when this video standard was introduced! (OLEDs can technically be driven interlaced but no controller does that in sync with NTSC as far as I know.) The best DVD players for this are from cca 2010 when USB was commonplace but they have no HDMI which could mean there is an internal scaler or advanced framebuffer (I know a device that recompresses its video output before it goes into the HDMI/composite output module: it’s horrible!). Speaking of codecs, some don’t support interlacing anymore (H.265 for example) so be careful.

Don’t use the converter if it cannot output 480i or at the very least 480p! Scaling should happen during playback, the files should be original resolution. You can also try non-trivial upscaling with some AI tools (best use them after the “computation-heavy” deinterlacing method, see above) but still DEFINITELY keep the original resolution file for archival.

use a [separate] worse quality VCR for cleaning

I don’t have experience with moldy tapes. It might be a good idea but adds wear; I’d just clean the VCR after every tape if I suspect mold. You’d still need to clean the cleaning VCR after every tape to avoid cross-contamination so it would be no easier.

Is [advanced deinterlacing] possible in OBS?

Idk, I just keep my files interlaced and stored as high-bitrate H.264 (I don’t have enough computing power to encode sufficiently good bitrate in better codecs). If I wanted deinterlacing, I could process the files with ffmpeg filters or some other tools.

What method do you recommend to [capture the video interlaced, preferrably as losslessly as possible]?

It’s been a while since I’ve done this but unless you’re recovering the Ark of the Covenant, it should be enough to follow these simple steps: use H.264 in OBS with high bitrate on a fast PC and preferrably using a USB 3.0+ port (even if the capture card is 2.0) to avoid clashing with other devices on the bandwidth-limited 2.0 bus. Check that the output is indeed interlacd. Look at stats/logs to see of any frames are dropped and investigate if it’s just the 59.94 Hz compensation, actual blank sections of tape or some part of the processing chain unable to keep up. Adjust audio levels; you might get better results using your PC’s mic socket rather than the capture card’s audio ADC (most tapes are mono anyway) but make sure to disable auto-gain or else quiet sections will get boosted like crazy, increasing the noise.