What's the POST endpoint to broadcast a transaction in copay's server? AFAIU, copay's server implementation is bitcore-wallet-service.

From its README, I can deduce that a multi-sig wallet will use the /v1/txproposals/ endpoint to create a new transaction proposal that the other co-signers will receive.

But, if the wallet is not multi-sig, what's the endpoint for broadcasting a transaction?
Multisig and singlesig wallets use the same flow. A Tx proposal need to be created in singlesig wallets also. Copay syncronize across devices, so you can have a non-signing copayers in a 1-1 wallet that submit proposals, for later approval.

The flow is:

create => publish => sign => broadcast.

You can check the official client https://github.com/bitpay/bitcore-wallet-client for details, and also the CLI client should be useful to see examples:
https://github.com/bitpay/bitcore-wallet