Firstly using a compressor to be sidechained and controlling the output of a delay is a very usable technique and it can be done. Gating is definitely not the correct effect to use in this situation.
The thing about using a compressor to duck anything is that it requires you to
spend some time setting the parameters of the compressor correctly and that is where people go wrong and they somehow expect it to be set perfectly and work perfectly the moment you insert the compressor. It won't work very well at first and requires some tweaking.
The
Threshold and Ratio determines how far down the compressor is going to duck. Sometimes it has to be set so the compressor ducks quite a lot before it becomes effective.
Attack determines how fast the ducking is going to occur once the key signal is present.
Release determines the time to which the ducked signal will be returned once the key signal goes away.
All the above posts are good for achieving what the OP wants to do but using a compressor to duck the delay is also very good too. Once you get it set up correctly it will sound very effective. It might save time later because once you do get it working nicely you don't have to worry about it much after that. A compressor is also good from the point of view that the ducked signal will be compressed something you cannot easily do other ways.
It is worth practising doing this because there are other situations where ducking can also be very effective. Ducking Reverb to maintain clarity with vocals is also another great thing to do. Also ducking the bass slightly every time a kick comes along. It improves the bass/kick clarity at that point a lot. But the settings here are quite different than the ducking delay/reverb we have been talking about so far though.