good question Tim. I always try to kill my catch as quick and humanely as possible.
I used different methods, depending on the type of fish.

For snapper and other species with bigger heads I use a spike to quickly stab their brain. 
That kills them really quickly. Then they go on ice or into the catch bag.

For fish with smaller heads or fish that need to be bled, like trout, mackerels, kahawai, etc I quickly cut through the gills and break the neck.

I'm not a fan of putting fish on ice without killing them first. It seems like a lot of unnecessary suffering to me. 

Depends on the fish.

If it is a fin fish (as opposed to shell fish, prawns, crabs etc) , I prefer to bleed them.

Most fish I keep I bleed. I cut the front of the gills where it meets the bottom of the jaw. Then bend the head back and snap the neck. Even though the fish is dead the heart will continue to pump for a few minutes and clear all the blood.

I don't know about the ice. You would need a pretty big boat to carry enough ice to make a difference. Ice alone wont work. You would need a slurry. So you would need enough ice and salt water to make a slurry big enough to make a difference, big enough to get down to -4 or -5 at least. I have never tried it, but I reckon a 40 lt esky would not get below zero. You would need a lot of ice to get the water below freezing. Then the thickness of the walls to keep the heat out...Maybe on a commercial fishing boat but just not practical for us.

all fish are different if you are trying to kill a carp they are quite hard to kill I have purchased a small baseball style bat from my local Tackle shop as I was told it is the only way to kill a carp quickly and 1 good smack on the noggin makes the fins relax instantly in Australia we are not allowed to put them back so some people leave them lying on the bank and i have heard of them still gasping two days later I find this unnecessarily cruel 

Flathead thrash around and can spike you really bad so I stun them (small baseball bat) then spike the brain with a sharpened screwdriver. The wider blade is easier to use than a pointed spike and if you miss the brain then you sever the spinal cord. I always put them in an esky with ice and cold water. They won't get below freezing but the cooler you can keep them, the better.

Here is an interesting read about dispatching and keeping fish fresh:

https://www.nzfishingworld.co.nz/latest/2015/08/iki-or-not-iki-question

I think everone is turning into store bought, hunters and fishers,its alright to eat fish fingers for an example which is trawled up by the tons at a time being crushed to death in a huge net or died in a crate or back of a ute going to the fish co-op,I have had a woman on the rocks one day try to release my fish in a keeper net i had caught that morning,saying i was cruel.When i fish i usually cut their throat and put in the ice box or keep them alive in a keeper net ,then cut their throats so fhat they bleed out,if theres a rock pool handy ile let them bleed out there ,then the first one i cut ,i scale gut and gill and back on ice,and i,m not about to become all touchy feely after 60 years of amature fishing and eating my catch