The point of proof-of-work is that the attack you describe, "transmitting a modified blockchain", is very, very expensive. Essentially any plausible block is "signed & trusted" by the very-difficult partial-hash-collision embedded within.
In this case, there's a single communications channel that could be spoofed by anyone with a transmitter powerful enough to drown out the legitimate signal.
You could stop sending new blocks, but that would alarm the endpoint because it will be stuck at block 100 (in our example).
The Proof-of-work scheme is pretty innovative.