I cant speak for anyone else, but this is what I do.
For general fish such as Roach, Tench, Bream etc. I would atempt to remove it using a barrel disgorger. I cant think of an instance where this has not worked. (Barbless hooks though)
For Perch I tend to have a very quick go with a disgorger if I think the hook is not too deep. If I cant get the hook in a couple of attempts, I cut the line as close to the hook as possible. I have found Perch can bleed VERY easily if hooked deep, therefore I prefer to cut the line quickly.
For carp, and I can honestly day I have never had a deep hooked carp, but i have had to rescue a couple from inexperienced anglers, I much prefer to cut the line. Although the hooks are a little bigger, so is the throat passage, and too tell the truth, the throat teeth make short work of the hooks anyway. One fish I dealt with at about 12 pounds is nor 20+ and fine.
For Pike, if you pull gently, most times the hooks will appear and u can remove them. I have caught pike with traces down there throats, but I am sure a treble hook can stitch up the throat preventing food being swallowed. Bolt croppers are useful for cutting hooks if need be. I must admit though, I use doubles or singles on dead baits now, and strike immediatly. Yes I miss the odd fish, but I cant think of a deep hooked fish in the last 15 years.
For Salmon, I cut off there heads and throw them up the bank, then take a knife and stab and stab until its well and truely dead...ooops...or am I talking about what the local salmon anglers do to my pike??..grrrrrr