Not really.

All the 120 vs. 60 setting really affects is your TV display refresh rate. The TV never extrapolates anything - any image interpolation is done in postproduction (when something is shot @ 24 fps and edited for broadcast in 59,94 fps).

What you described is rather the difference between a show shot in interlaced video @ 29,97 fps (59,94 fps) where half the image (every other scanline) is displayed first, then the other half (how this is handled by the screen is what the 120 vs. 60 setting affects) versus footage shot in progressive video (23,976 fps, aka 24p) where both halves of a frame are identical (aka full frame video), except 1 frame every 4, which is interpolated (an automatic merging of the frame before and after) and interlaced. It's called "pulldown" in the trade, and its purpose is to convert 24 fps to 30 fps).

Picture looks sharper in 24p (especially when paused), but movements, especially fast ones, look far more fluid in 59,94 fps. Most high scale fiction is shot in 24p (just like movies), edited and finished in 24p, then transferred with added pulldown for 59,94 fps interlaced broadcast. A great deal of TV shows (live stuff, non fuiction and so on) is shot directly in 59,94 fps. This is what your 60 vs. 120 setting is for.

The picture looking too sharp/too clean has nothing to do with that, it's just a matter of getting used to HD definition and look (and the brightness etc.), especially acute for people who switch from SD to full HD. After a while, you get used to it and the old SD TV look like crap.

In post, we dirty up the image (add noise, crunch the colors etc.) to avoid the "too sharp" effect, but if people have a less than ideal color set up on their TV, it makes things worse instead.