I can't find it in their show notes, so maybe I misremembered the source. Sorry.
In a more wide ranging search (flu and COVID same arm), I found this. As far as I can tell, at least an inch apart will help distinguish between local reactions, separate arms might lead to stiffness in both arms which might be more desirable than completely taking one arm out of commission. The only mention of a downside to same arm is that "high dose" flu shot in the same arm as COVID might produce more discomfort than just a "doubling up".
Everything seems to be in agreement that effectiveness is not helped or hurt either way.
Again, my apologies for not getting this right.
On the plus side, now you have an excuse to subscribe to TWIV and browse their back catalogue. It really is a great podcast.
... is pretty darn common across the board. I've never done any government contracting but I find it hard to imagine that it could be worse than the private sector.