Vintage sound. In small amounts, it can add warmth. Exciters do similar things.As I do.
"usually" is the key word here. I rarely use exciters either.It is only a matter of level. The hotter, the more harmonics you'll get due to saturation. You certainly can wreck any tape recording if you go too hot.
The same thing applies with the compressor method, ie. if the curve does not match the input level, this doesn't work as intended.
The advantage of the compressor method is that it doesn't jitter like tape. Thus, you can add a high pass filter to basically get the harmonics only and mix them with the input. This can help old tape recordings which are missing the high end to a degree. If you just seek vintage sound, the tape method should do, of course.
While tape saturation (and imitations thereof) add 3rd harmonics, you can also try a ring modulator, which can produce 2nd harmonics (with twice the dynamics, so you probably want to compress them). Also works great, if you're restoring old recordings.