A Blockchain that merges mining forks and could be more power-efficient?
Something that is very similar to what you describe is the concept of "Uncle Blocks" used in Ethereum. Even there, though, there are no partial blocks and nobody would continue to mine on a "half-finished" block. Ethereum has significantly shorter block intervals which causes multiple blocks at the same height to be found often and allowing to reference them as uncle blocks mitigates the waste of that to a degree. Introducing progress into mining is problematic, because it biases mining success towards miners with more hashrate.