Now there is a "winner takes all" game when mining new blocks. Is it possible to mine transactions instead? I mean, generating as many zero bits not in block, but in single transactions, then counting all these leading zero bits in total and have required block difficulty based on that. Then, the block reward will not be sent to just one winner, but splitted between all people mining their transactions.
A pool has the effect that you are looking for. The block reward is split among all of the members of the pool.
BTW, the process of counting zero bits was mentioned in the white paper, but it was never implemented that way.