Re: It's getting annoying..
David, it sure looks like you have been trying to find solutions, and I salute you for that.
I am wondering .... if you need to increase student's ability to take ukemi and you need a bit of a workout...why not spend the first 15 minutes of a class specifically on ukemi skills, and then close the class with line practice jiyuwaza - you start as nage - they line up and one at a time (not randori style) each student comes as quickly as he/she is comfortable with either a designated attack or any attack and you do simple "randori throw" paced to each one's abilities to let them get more comfortable....if they each run up to take their turn you will get some exercise. Then they get to practice randori throw with you in line as one of the ukes.
You could of course do technique on them when you are nage but in my experience for newbies it makes more sense when they are nage to be limited to the simplest of kokyu throws.
Good to teach ukemi skills (including giving a real attack while moving, rather than just hurling oneself), pacing, flow, etc - and good way for everybody to work up a bit of a sweat.