Polonius makes it so clear that this hiding behind the arras business (the one that gets him killed) was 100% not his idea. He flatters Claudius about it but there’s a quality of the Polonius Doth protest too much about it for me.
I’m curious, too, about why Shakespeare makes it so explicit that this is Claudius’ idea to eavesdrop on Gertrude and Hamlet. It feels like it serves to magnify his guilt, like it shifts the responsibility for Polonius’ death a TINY BIT off of Hamlet’s shoulders.
I suppose it also supports Hamlet’s idea that it would be Claudius behind the arras. This variety of spying being so clearly more Claudius’ style – given that he came up with it.