Canada

Except that that opens an even larger can of worms.

Currently, the GG is selected on the PM's recommendation. We've gotten away with that so far because there's a disinterested party staring over the PM's shoulder in the form of the monarch (reducing the chance of really dodgy recommendations) and because no PM has yet run off the rails the way Trump is doing down south.

In every government decision except the selection of the GG, the GG is the disinterested person staring over the PM's shoulder. Even if they don't normally exercise any power, I don't want a position that could act as a check for the PM being decided on by the PM. So we then have to move to some other method of selecting the GG. The most usual method in other countries is by holding a separate election, but that immediately pisses a huge amount of money down the drain. And that's without dragging in the constitutional amendment considerations.

I'd rather just spend a trivial (on national budget scales) amount of money on the monarchy and keep the worms firmly enclosed in their cylindrical metal containers, thanks very much.