One of my coffee-table books on Austin Healeys had this story. You can take it for what it's worth, considering the source.
The quarter-elliptic springs and lack of a trunk lid had more of a reason than just saving money on those components. It allowed the rear of the car to be adequately stiff without much structure. Obviously, with a half-elliptic spring, you would need something strong to attach it to, and also the hole in the back created by a trunk lid would require extra stiffening. And, as you state, the quater-elliptic spring had some disadvantages, such as lateral support.
But the lack of a trunk lid was a real turn-off for a lot of potential customers, more so than originally surmised. So, the Sprite got a real trunk, which required more structure, and from there it was a relatively small step to add the full springs and necessary structure to support them.
I think this makes sense mechanically, but whether it is true historically is a separate question.