Prematurely broadcasted timelocked transactions are invalid and ignored by the network. That's why additional application logic is needed to broadcast the timelocked transaction after the timelock has passed.
They can't be included in a block until the timelock is reached, but I do assume that they stay in the mempool for a while.
Still, it makes since that you should keep your wallet running at least until few days before you die, so it doesn't disappear from the mempool. Most wallets, including Bitcoin Core, will keep broadcasting your transaction until it is included in a block.
If I recall correctly nodes usually drop transactions off their mempool within 3-4 days or so. Maybe after a bit longer, but definitely a timeframe that's too short to be practical for a dead man's switch. That is assuming a not-yet-spendable transaction is kept around in the first place.
Good point about wallets keeping rebroadcasting transactions. In the case of a dead man's switch I personally would probably double and triple check that the wallet does indeed keep rebroadcasting the transaction but if it does you could keep the surrounding application logic at a minimum (if additional logic is even necessary at all).