Why wouldn't sending and receiving over the same frequencies work?

Gigabit Ethernet works similarly to what you proposed. Each of the 4 twisted pairs is simultaneously bidirectional by each transceiver subtracting its own signal from what's on the wire.

I note that your proposal only works for diplexing (full duplex, 2-way multiplexing), not n-way multiplexing where n > 2.

DOCSIS can't work that way for two reasons. First, because everyone in your neighborhood is on the same wire, so n > 2. Second, DOCSIS HFC networks were built for cable TV and have amps that only amplify frequencies below ~50-200MHz in the upstream direction, and only amplify frequencies above that "split frequency" in the downstream direction. So you'd have to do massive infrastructure upgrades to make it so all frequencies are equally amplified in both directions.

I suppose DSL could work like you propose, but I imagine there are other hassles and constraints of that unshielded, unbalanced medium that makes it better to use the frequency division multiplexing it uses today.