I think rationing is a bit of a dubious word to describe it. In both public and private healthcare you have limited healthcare resources split across the public so "rationing" occurs either way. The real difference is that in public healthcare it's rationed by the severity of your condition while on private it's rationed by the size of your wallet.

If a private system would've allowed your sister to get treated faster, then it would likely mean that whoever was ahead of her in the queue for public health treatment (and thus had a more severe condition), is now losing out on treatment instead