I don't think I would emphasize the "strong/rigid - element" in our hanmi. We don't try to "hold the ground" by using a certain stance/kamae.
It' more about connecting with aite, and leading one's own and also his feeling down, using this connection.
Do you sometimes do a technique just standing on one foot? You don't need a wide stance to be centered and grounded.
I am not sure you understand me correctly. What I am trying to say is that you can physically, regardless of Aikido, stand wide, which is strong, but rigid. Or you can stand narrow, but agile. In Aikido you find the balance between the two that fits your physique the best. Good starting point is a stance at about shoulderwidth.
I cannot think of any technique on one foot, but we do have techniques we stance is very narrow (e.g. koshinage and in some weapon work).
Agreed. Your stance should be neutral, in balance. The tests you propose are very good.
We often refer to the third point: points of unbalance. Two exist: one in front of you and one at your back. Imagine a line going through your front foot and rear. Now image a line perpendicular to it that goes through your center. This line is where you are weakest (unbalance) and you cannot divert forces downward. Bit technical but hopefully makes sense.
I use this in class to explain the 'no force' in Aikido. In every technique you attempt to direct aite in that direction, keeping him/her off balance.