Lastly, given that it has to be kept secret to the beneficiary, the third-party has to keep the necessary information to broadcast or spend instead.
Which in my opinion creates a greater security issue, because normally, the beneficiary is someone we know. Therefore, if we have to give a time-locked transaction to someone, trusting that they will indeed broadcast it when the time comes, wouldn't it be better to rely on our beneficiary instead of an intermediary? This is purely subjective, I am just expressing my thoughts. And lastly, a time-locked transaction will send coins to an address. How would we make sure that the beneficiary owns the private key to unlock the coins at that address?
What I want to say is, in general, wouldn't it be better if we let our beneficiary now about our intentions and plans?