Is the mempool stored on the blockchain? If it is, isn't it then easily possible to have all wallets scan the mempool in order to give immediate updates that a trx was actually sent? Just knowing that a trx is actually sent is a big deal for a receiver rather than have to wait for the tx to be included in a block first which can take some time if the mempool is congested, or if a block takes a long time to be mined.

Actual confirmations is another subject and not what I'm asking.