Why would a Bitcoin miner take less than the full reward?
Likely you're correct, it's not useful in practice. People could, however, use value destruction as an costly signal. 

Voting example; I run a poll of mining pools; number of Satoshis less than the full block reward, up to 5, is the vote for which of 5 proposed algorithms to run in the future. Once a majority of blocks over some time period vote for option X, everyone agrees to switch protocols. (This could allow identityless voting by miners.)

Commitment example; A miner can "promise" something, and commit the value destroyed as a signal that they are willing to do it. If a miner wants to extort the network, such commitment could be effective in showing that they are serious.