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.
If the client has block 5 and its balance sheet he can verify both block 6 and block 4 by fast-forwarding or rewinding the new/old transactions on the balance sheet. Verification in this case means "if the beginning of the chain is verified then so are all the nodes up until the end."
The client may end up with multiple long chains, which will either be connected in some way or one will be an attacker's chain. The client should be able to request older blocks until the point where the chains connect or the attacking chain is overpowered.
I think you get too much fragmentation if you only get new blocks, but the idea of "mostly broadcast" may still be a good aim, given that the number of old blocks required to remove all doubt may be quite few.