The problem with automating ‘mute’ of an FX channel is that it is being sent the signal previous to where the OP wants the delay. That will have remaining delay from previous audio on the track because it is always sending to the delay.
Automating the ‘power’ button on the ‘send’ for that track works better when wanting exact placement send to an FX channel. And that also takes no more resources.
Just hit the ‘W’ on the channel you want delay from, and then the power button for the delay send where you want it. If you get it wrong just edit the automation on the track in the project window. Simple.
Every user has different ways of doing things, and all are correct. I tend to do multiples of such delay addition to end phrases in my mixes so copying tracks is not the best way for me. BTW, I find that also sending the delay FX channel send to the vocal reverb makes the delay more pleasing. Just me tho…