how would it work so say "hamesha nahin" rather than "kabhi nahin"?
I would take it to mean 'Peter doesn't play always', ie, he does something other than playing every now and then.
Think of it as a logic statement.
Kabhi is logic statement meaning "sometimes" meaning "at least once" the opposite would be never
Why is "पीटर कभी खेलता नहीं है" wrong?
That is also correct. You can report if you see the sentence again.