Rarely, Seldom, or Occasionally in my view equate to something happening 'less often than it does not happen' as you put it.
They do however connote varying degrees of frequency, of which rarely seems the least often and occasionally somewhat more frequent.
Other similar phrases include...
'from time to time'
'every now and then'
or just 'sometimes' ...would connote to me something happening less than 50% of the time
I don't think there is an exact opposite phrase of what you're looking for.
if you really need to be specific about something usually not happening you could just adjust the latter part of the sentence.
eg. "More often than not the birds fly away"
The opposite is "More often than not the birds don't fly away"
Depending on the sentence it could sound strange but is a potential solution.