] But I just noticed last week that in the two "big" eavesdropping scenes (snip) it is Polonius, and not Claudius, who makes the suggestion that they participate in trying to overhear the truth.

I am beginning to think that Polonius is a lot more in control of the whole situation than perhaps meets the eye. Esp. when it comes to Claudius. Or perhaps, Claudius is a bit of a puppet?
Claudius is obviously (to me) very reliant on Polonius. However, Rosencrantz & Guildenstern (or is it Guildenstern & Rosencrantz?) are Claudius' idea?

This could also hearken back to my theory that Claudius is not a thinking man, he is a man of action. These ideas wouldn't occur to him by naturally. Polonius is a proven spy - as documented by sending the spy to temp Laertes. Gives a good reason, plotwise, to have that scene in the play at all.