The plate is the full length of the shoe. Not sure how you're imagining adding springs in a way that is more conducive to running. Keep in mind shoe weight matters quite a bit.
Sure. I think Nike is probably aware of this and took it into account when designing this shoe. If you have a better design (your earlier comment suggests "just add[ing] springs to the entire shoe"), there's probably a financial motive for you to figure out how to sell that. (I think you are probably understating the difficulty of designing a better shoe, for any definition of "better.")
It's possible to run at or near an elite level without modern running shoes. For example the 1960 Olympic marathon was won by a barefoot competitor (his winning time of 2:15:16.2 would be considered sub-elite today) [1].
One can speculate why all elite runners use shoes. Worth noting that running shoes designed for racing, as opposed to training, tend to be thinner and lighter and closer to a simple sandal [2]. I personally reverted from bare feet back to shoes after the occasional thorn or bit of glass getting stuck in my foot became too annoying.
> they have a carbon-fiber plate in the midsole, which stores and releases energy with each stride
... the achilles tendon + calf already play this role.
Just don't heel strike.