OK, I see what you are saying. You are effectively suggesting pipelining the broadcasts of the update transactions. I think this introduces a problem that a node in the circuit that withholds the preimage for too long can force all upstream channels to be closed, at only the expense of their one upstream channel being closed. I believe such an attack could significantly disrupt the network.
Let me elaborate on the way I'm thinking about this:
So say I'm a routing node with an upstream HTLC with CLTV = X. I need to ensure that if I learn the preimage, that I have time to broadcast and confirm an HTLC-success transaction before height X. We'll call this number of blocks D_success. So if I know the preimage, let's say X - D_success is the latest height that I can safely broadcast the HTLC-success transaction, assuming the settlement transaction is already final (ie. the update transaction is confirmed and the CSV delay has passed). So now I also need to know when to close the channel with the update transaction. I'll assume it will take at most D_update blocks from the time I broadcast the update transaction for it to be mined. So unless the downstream HTLC is already failed, I should always close the upstream channel at height X - D_success - CSV_update - D_update.
Now we'll look at the downstream HTLC with CLTV = Y. In order to minimize the safe delta between the upstream and downstream CLTVs, I will want to broadcast and confirm an HTLC-timeout transaction as soon after height Y as possible. So assuming that the downstream settlement transaction is final at height Y and it takes at most D_timeout blocks for the HTLC timeout transaction to confirm once it is final assuming no double spends, then Y + D_timeout is very latest I might learn the payment preimage from the downstream channel on-chain. So I should be safe as long as X - D_success > Y + D_timeout. This assumes that the update transaction for the downstream channel is already mined and the CSV has passed. However, we know from above that I had to close the upstream channel at time X - D_success - CSV_update - D_update, which may very well be before Y. So if the downstream hop waits until just before Y to publish the preimage, they can force me to close my upstream channel. This applies transitively for further upstream hops, assuming a large enough CSV value.
Granted, upstream hops can watch the blockchain for preimage reveals in other closings transaction and perhaps fulfill off-chain if there is sufficient time. This would not be possible with payment decorrelation through scriptless scripts or the like.
Does that logic sound right to you?