A notification system should work to minimize the work for Alice to check every week/day if the address was funded.
But still doesn't totally solve the "fee" and "complexity" problem since Bob can broadcast the tx anytime
that will require Alice to make another Time Lock script (another transaction to transfer the funds) in order not to expose the redeem script and public keys for privacy (or future security) reasons.
This partially breaks the "trustless" system.
Redeem script and public keys of Alice are exposed to Bob once the transaction is transferred to him. Privacy broken here for all Alice addresses involved in the transaction (at least Bob knows them now)
Yes if Bob broadcasts before accident, Alice will withdraw from the lock script to ANY output under her control. Even if Alice only suspects that Bob is going to play dishonest game she will move funds to other outputs.
At this point Alice funds are absolutely safe from Bob (tx can't spend outputs twice) although are not partially private. It is now up to Alice to decide what to do next: create another inheritance transaction to Bob, break the relationships with Bob, involve another heir, etc.
Notification system is something that can work on Alice mobile or even can be checked by Alice manually in some simple cases. Say, if lock period is 3 months, check once a months is enough. Either way checks can do not discover Alice addresses to the world.
I don't know what you mean by "
trustless" system exactly, however, I see that: privacy is partially broken - YES, any kind of intermediary in the process - NO