Of cause it does. Think about it. With Local On engaged, the sound generating circuitry receives two note-on instructions for every note your press; one from the keyboard and another that has gone through your computer. The last one hast taken the long way around and arrives a bit later then the first one. This latency is the cause of phasing you hear. It also uses twice as many voices per key, thereby halving your available polyphony.
This problem was the reason why local on/off was invented, some 30 years ago, in the first place.
Set your Kronos to “Local Off”, while you’re recording, and this will solve your problem.
Another option is to deactivate “MIDI Thru Active” in Preferences/MIDI. This will also prevent duplicate key-on messages. I prefer the Local-Off alternative, but I guess that’s a matter of taste.