In their words, “the only time we're allowed to pester them”.
Imagine if the boot was on the other foot. Imagine if they had to ask us more than once to do their work for us. Imagine if they had to pester us to literally do our job. We'd be sacked.
Seems to me that they're covering for any potential failure in designating work, and letting the rest of us take the blame.
Also, this is exactly how a company culture is developed. Only those who are persistent, extroverted and disagreeable will get anywhere in this environment. The quiet and introverted among us will be set up to fail.