This seems analogous to giving up the belief that free will is somehow ontologically basic. The experience of having made a "choice" can be arbitrarily superimposed on pretty much any action I perform. I value the experience of choosing, but recognize it as subjective fiction. Similarly, I find your suggestion

perhaps it can run some computations on the side to generate the qualia of anticipation and continuity

to be highly intuitively acceptable, but I feel that I'm missing something, perhaps a compelling counter-example.