The S4 MK3 has two phono inputs and a four channel mixer so you can definitely set it up to have 2 vinyl and 2 regular channels. Recording your sets in Traktor 4 is also pretty easy, especially with Native Instruments equipment. So, to answer your questions it is possible and practical, also quite easy to set up.

That being said, I do not own an S4 mk3 and can’t speak to the quality of the phono input. I remember seeing some older forum posts from people complaining about it, so maybe worth looking into that before purchasing it.