TL;DR: Why are transactions (240 bytes) added in the block body, instead of the SHA256 hash (256 bits) of the transaction?

From what I understand, a transaction (approx 240 bytes) is 'broadcasted' to the everyone in the network through inv-messages. A miner collects transactions, and adds 1 or more in a block(body). After finding the hash, a miner 'broadcasts the block (header and body). In essence, transactions are send twice. First the actual transaction, second, a cumulated set of transactions.

If a transactions is broadcasted and everyone receives the transaction, why not add a hash (256 bits) of the transaction in a block body? Propagation time of a block would be similar, the number of transactions send in a block is increased by a factor 8-ish.

One argument could be that not all mempools are synced. However, assuming that a transactions are broadcasted much faster than a block, part of the ruleset could be that a miner only can create a block from  transactions received. In practise, if a miner sends a block to its direct peers (in the P2P network) and the transaction is not known by these nodes, than the block is not valid. Since a block creation time is (on average) 10 minutes, transactions added to the block have plenty of time to propagate.