Thanks. You paraphrased my question.
What does it come down to when necessity raises its ugly head?
It may be because of this: take part (in something) = participate (in something) participate doesn't take an object; however, if you wanted to apply one, it'd be in the form of a prepositional phrase (in something); the in is the inseparable part of the prepositional phrase, but not of the verb participate (or the phrase take part), hence it is not required while referring back to an activity in question.
get rid of something = discard something discard requires an object, and so does the phrase get rid of, even when referring back to something already mentioned; the of is the inseparable part of the whole phrase get rid of.
This is only my interpretation of the matter under discussion. You ought not to take it for granted.