Participant, no, but a manager of the time and the response, I'd say yes.
But I have to admit that I've not been part of a debate team to know the rules.
Well the tv "debates" are not like scored college debates. The format is way different.
In NCAA debating you are arguing for or against a single proposition, and the format is like you would see it on your ballot --
- argument in favor
- rebuttal of argument in favor
- argument opposed
- rebuttal of argument opposed.
On the tv debates, they are debating a number of issues, and the format is simply as follows usually --
- statement by the 1st candidate
- rebuttal by the 2nd candidate.
In either case, the moderator or judge is not supposed to say anything, other than calling time when expired.
In an NCAA debate, the moderator is also the judge who scores the debate.
On tv, it is the audience who will form their own impressions, and then afterwards there will be news commentary and editorial as to who looked and sounded good and therefore "won the debate."
So as you say, managing the time -- clearly yes.
Managing the responses -- absolutely not.