That last bit is the thing. Equivalence is all about making photos that look the same (on the basis that a photographer has some intention about how the photo should look). The look of a photo, in terms of motion blur imbued by a particular exposure time, doesn't change with frame size. The look, in terms of DOF at a selected f-number does change, meaning that you'd have to change the f-number to get the desired DoF with a different size sensor. Likewise the look in terms of AoV for a given FL does change, so if you want a particular AoV you have to change FL with sensor size.
The ISO thing is simply because, as you say, changing the f-number with the same shutter speed and scene illuminance changes the exposure, so if you want the photo rendered with an intended lightness you'll need to change the ISO.
The question that raises is why is exposure important? This is a genuine question, but when I ask it I tend to get answers such as 'if you don't even know that, there's no point even discussing with you'.