Yes, this is effectively what 4DVar data assimilation is [1]. But it's very, very expensive to continually run new forecasts with re-assimilated state estimates. Actually, one of the _biggest_ impacts that models like GraphCast might have is providing a way to do exactly this - rapidly re-running the forecast in response to updated initial conditions. By tracking changes in the model evolution over subsequent re-initializations like this, one could might be able to better quantify expected forecast uncertainty, even moreso than just by running large ensembles.

Expect lots of R&D in this area over the next two years...

[1]: https://www.ecmwf.int/en/about/media-centre/news/2022/25-yea...