Bitcoin can already work like this, just not as an absolute. This is similar to my own 'datacasting' concept.
What is your datacasting concept?
Use the search function. Look for 'datacasting' and 'Dash7'
One reason why I think bitcoin can't work like this is that a client who only saw new blocks wouldn't be able to know whether the transactions contained within were valid or not. This seems to require balance sheets...
that is the chicken and egg problem, isn't it? A client that uses balance sheets cannot independently verify transactions based on blocks prior to it's own initial startup, but a client that uses the full blockchain to verify transaction history doesn't need balance sheets.