Will nodes configured with a larger-than-default mempool automatically retransmit transactions that have been dropped from the default mempool?
So far, Bitcoin Core rebroadcasting is a wallet-level responsibility. Transactions that aren't relevant to any of the Bitcoin Core's internal wallet are not rebroadcast at all.
There is work to change that: https://github.com/bitcoin/bitcoin/pull/21061. This makes sense, as right now, observing a rebroadcast is a very clear sign that the transaction belongs to the wallet, which is a privacy leak.
So I think in short the answer is: yes, it makes sense, but so far it hasn't been done.