There are two issues with a system like that
a) Bitcoin has no mechanism for detecting the first broadcaster.  Say you hold the tx for five years, and then you broadcast it, the miner including it in a block could simply take credit.  There may be a novel way to avoid this but it isn't a solved problem.  The reverse is also true.  A node could collect a fee in advance but then could simply delete the tx without loss.

b) The fee system is designed to prevent attacks which degrade the performance of the network.  With a finalized tx the tx will eventually end up in a block and thus the network can be assured the fee means a real cost to the user.  An attacker trying to degrade the network would need to a pay a steep fee to do so.   With locktime transactions the fee doesn't guarantee a cost.   A user could spam the network with millions, potentially billions of future txs all with hefty fee and then end up paying for none of them by creating another tx prior to the locktime and broadcasting that making all those held txs invalid and thus the fees promised, never paid.

Still Locktime is a useful tool even when the tx is not relayed.  I will give just one example.  Say I operated a "green address' type site.  You have 1 key, I have 1 key.  The address requires 2 of 2.  I can't spend your coins without your permission and if a merchant trusts me to not approve a double spend then they can trust 0-confirm payments from you.  That would be a very useful service but while I can't steal your funds (2 of 2 remember) I can hold them hostage.  This could be for malicious reasons, governmental interference, or just due to incompetence (if I lose my key your funds are lost forever).  LockTime can be used to end that "stalemate", after each transaction I provide you a future locked tx which sends your coins back to an address you control.   If either one of us loses our key or refuses to sign a tx eventually that locktimed tx will become finalized and you can simply broadcast it to the network to recover your coins.  That LockedTimed tx acts like a "voucher" to recover all your coins in the event of a problem but one which can only be used in the future.  That is just one example but it shows how LockTimed tx don't need to be broadcast to the network prior to their LockTime to be useful.  Other nodes don't need to hold the locked tx because you have a very strong incentive to hold your own locked tx (i.e. in this case it lets you recover your coins).