Upgrading to a dif Toilet, Flange is stuck???

Jeff I like the lags better myself but some times whoever cut out the plywood for the toilet flange cuts a little to big and the lags have not enough material to get a good bite. Or the plywood is rotted from a leak over the years. From your pictures it looks like your plywood is dry and tight around the flange. Personally I would not be pulling or twisting that flange around.
That picture I posted of the repair flange yes it would just slide under the flange then screw the flange down. I don't know why they show that nut on the johni bolt in the pic that would be the nut ontop of the toilet with a washer.