Does a blockchain need a distributed hash table?
No. Peer discovery does not use a distributed hash table and cannot use a DHT. Once you are connected to the network, there is no need for you to find more peers to connect to (and most nodes have a limit of 8 anyways as that is Core's outbound connection limit).

The broadcasting of transactions and blocks themselves does not rely on the broadcaster to find all peers and send the data to each peer. Rather it simply sends the transaction or block to the peers it is connected to, and then those peers check the block and transaction, and if it is valid, they will send the block or transaction to their peers, and so on and so forth.