I had a discussion yesterday whether "Succumb to the Cold" would leave one focus token on a Elite-Character, when it only has one focus token to remove, or not.

My assumption was: yes

What do you guys think?

you are correct. The event card is played as a reaction to a unit removing a focus token (or in this case refreshing) so regardless if the unit is elite or not the unit cannot refresh again in the same turn. Therefor, a focus token is placed back on the unit.

Concur. Succumb > Elite. Also, pro tip, don't forget it can be used ANYTIME a focus token is removed (like with Trust Your Feelings or Wedge or Leia).