You're basically just describing fee-less digital currency. The problem is that achieving the fee-less part by using a blockchain makes some extremely undesirable tradeoffs. To name a few:

  -Money laundering becomes impossible to prevent when no entity has control over identity verification of the users. 
  
  -All transactions are public, so normal users who won't be trying to lie about their identity lose all financial privacy.
  
  -Undoing transactions due to fraud or simple error becomes extremely cumbersome and ultimately requires re-centralization of authority that crypto supposedly was created to avoid.

Digital fiat transactions operating in the current financial system are the best of all worlds, and unsurprisingly they are already extremely common.