additionally if the server gets hacked, the hacker can broadcast the transaction just to be annoying and you will have the same problem.
and finally if you use locktime in that transaction (which is usually the suggestion) the bigger problem is that you have to keep accessing your cold storage on those intervals move the funds to invalidate the previous tx and create a new transaction with a new locktime.
Well, the difference between this method and trusting your private key to some third party is that here the worst case is your family members getting coins prematurely, whether with third party they can just take your coins at any time. As far as I know, there's no perfect scheme that doesn't have some risks or tradeoffs, but feel free to correct me if you know some.