60 is just slightly higher than the limit of what we can notice, if we just stare at one spot and look for the stutter. But if we're following motion across the screen, our visual system tracks smoothly while an unsmoothed object jumps positions. It takes an insane framerate to eliminate that effect, so it's usually just reduced instead of eliminated. Mostly sports and immersive games care about that; everything else, not so much.
(Personally, I wonder what happened to Chuck Jones' "smear animation" technique, which used about 3 frames of 24fps to move a character a long way, smoothly. Why don't more games do that? Anyway...)
The illusion of motion works at an arbitrarily low framerate, as evidenced by any 2-frame animation. But different rates have a different "feel" to them, depending on how "jerky" they are. 24 is fast enough to feel like real life, but still has a sort-of "aloof" quality to it, that works well for movies. Subconscious suspension of disbelief.
From a mathematical/technical perspective, it really messes with things if the framerates don't match, but you might not notice that mess in the result:
- An integer mismatch is quick and easy, and still looks good: just duplicate or drop frames as necessary.
- 60/30, for example, reduces to 2/1 = 2.
- A rational mismatch takes a little bit more work, and the less-work versions can easily become noticeable: have two different numbers of duplicated or dropped frames, and alternate between them to keep the timelines aligned.
- Alternate between 1 and 2 dropped or duplicated, for a ratio of 5/2, which is the same as 60/24.
- For 30/24, which reduces to 5/4, it's mostly frame-for-frame, but every 6th frame or so is dropped or duplicated.
- A very close but not quite match, like 30/29.97, is probably not noticeable even for a quick-and-dirty method that drops or duplicates 1 frame every 33.3 seconds or so.
- Or depending on what's happening at that exact moment, maybe some people do see a glitch, and pass it off as "a weird technical anomaly".
(I might be slightly off on the exact numbers, just doing this in my head, but you get the idea.)
If the one thing you can't change is 24, and that looks fine, then I'd say to set all of your gear to 24 to match it, including the camera, OBS, and everything else in the chain. See what that looks like. If you like the "cinematic effect" on you too, then keep it. If not, bump yourself up and see what the up-converted other stuff looks like. If that's okay, keep that.
