You are right, “chybět” is construed in a way that's hard to imitate in English. You might think about one of the following (extremely) crude translations:
- to me, František is not missing
- to me, František is not lacking
- for me, František is a good riddance
That's why František is the subject of the Czech phrase (in the nominative) and “mně” (to me) translates to “I”.
But of course the elegant and correct English translation is the one given in the exercise.
Seeing that you have level 23 in Spanish: “chybět” is a bit similar to the Spanish “faltar.” You could say “František no me falta.” In contrary, the English “to miss” is more similar to the Spanish “extrañar.”