Is it possible to build mining reward pooling directly into Bitcoin?
I believe this would be very difficult to implement in Bitcoin.  Not only because of technical feasibility, but also because gaining concensus would be difficult.


  The mining reward could then be distributed in one transaction
  proportionally to all miners who have found hashes.


This appears to be making modifications to the coinbase transaction, which would almost certainly require a hard fork.  Also, what is preventing the miner who created the block at full difficulty from simply cutting out all the miners who submitted blocks at n-1, n-2, etc...?  How would other network participants know that other miners have been cut out, so know they should reject the block?  That trust model would have to be solved.

If a hard fork is necessary, you would need concensus from the mining pools...which are the entities your are attempting to disrupt.  Even though their hashing power comes from individual participants in the pools, it would be hard to push something through that isn't necessarily in their best interest.

Finally, what would prevent individual miners from banding together to "game" the new protocol, thus re-creating mining pools on top of this mining pool protocol?  Building pools into the protocol won't stop people from pooling together...it will just give them different goals for them to pool resources for.