There has been a lot of discussion about sigmoids, especially tanh. You can search for it, but a few years ago (when I was a lot more active and had time) I compiled a kind of list.
https://forum.cockos.com/showthread.php?t=190087
My favorite subtle saturation is Loser/Saturation. It's subtle, it's sweet, you can cascade it, I really really really like it. If you're going to go the route of analog summing you want subtle, not obvious. This is one way to do it and it's adjustable. I'm almost positive that I figured out a more generalized way to do it that didn't rely on calling sin()... but it's apparently been years and I just don't remember.
Argitoth's post about sigmoids is definitely one to look at. If you like long-winded discussions about math you can read some of my posts, I tried to explain some of my reasoning a few different times in a few different posts.
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If I was to attempt this style of saturator I would probably use a compression-style waveshaper with a soft knee, e.g. a ratio of maybe 2:1 and a super wide knee, like from -15 or -20 to 15 or some such. Would probably want to have the saturation in parallel, probably filter it as well. You can find a post that I wrote on soft knees using a quadratic spline in that list above. Again, probably overly long-winded, but sometimes you get excited about things.