I very much felt that way when I started, and for me it got much easier when I gained confidence just through exposure and practice. The first 30 cats who screamed bloody murder scared the crap out of me...now I'm much more "yeah yeah you're gonna kill me, I get it."
What also helped was working with wildlife. I'm now much more afraid of raccoons than cats.
For extremely fractious cats I pretty much have my routine down: have everything set up for whatever we're doing, have a minimum of two thick towels, snap collar or muzzle, go into a closed quiet room, and get 'er done. Once you manage to do a few dozen it becomes a necessary evil instead of something I dread.
If you can get them with thick towels, a muzzle, or just a quiet room, that's not an "extremely" fractious cat.
That's just a resistant one.
We routinely have such fractious cats that you can't even consider opening the kennel door on them, much less opening it to get a towel or muzzle in there. That's extremely fractious. And it's what propofol is for.
Fortunately, they don't usually get that way until they've been in the hospital a while, so you've had a chance to get a line in them first back when you could wrangle them with a towel. I've had cats we've literally propofol'd for every medical progress exam, TPR, etc.
On the positive side, it helps get them eating.
Ideally, you'd be 'training' them during their hospitalization so they'd get more compliant rather than less. But
Working with wildlife does help improve your reaction skills, no doubt. Squirrels scare the hell out of me. Stupid little chittering deadly teeth puff tails.
I agree with
@dyachei - restrain it like you mean it. Often it will actually help the animal (in addition to just being a more stable platform for the doctor or other tech to work on the animal). One bit of caution with it, though - if you have a hospitalized animal, jumping to very firm restraint can make your problem worse as hospitalization goes on because they'll learn to interpret the cage door opening as "I'm about to get squeezed." For hospitalized cases it's sometimes better to just take what you can get and maybe forego that q4 temperature or whatever. Or use chemical restraint. I get super annoyed when I come take over a fractious blocked cat case and I see q4 temperatures on the chart .... I mean, really? It's a FLUTD case. I don't really care about its temp too much. And I don't need to be stressing the cat out to great extremes every four hours given that stress is probably playing some role in its urinary issues that we're trying to resolve. So be smart and only get data that you actually need and is actionable.
My approach in the exam room - and I think everyone has to develop their own - is that I go in, sit down on the floor, and start talking to the owner. I get my history and whatnot sitting on the floor pretty much completely ignoring the animal. Gives them time to get used to me. I'm down low where I'm not a threat. The majority of the time, even a timid or fearful animal will have come over to greet me on its own terms by the time I need to examine it. Obviously I don't do this with the 100-lb dog that lunges at me the second I walk in the room. But I usually get more time in the exam room than in a typical day practice, so I can take the slower approach 80% of the time. In a busy 15-minute-appts-all-day-long setting you don't get that luxury. I have had multiple owners comment along the lines of "you're the first vet I know who has sat on the floor to examine my dog." I take what I can get from the dog for my exam and while I try to stick to my 'normal' exam routine, if it's not working for the dog I change it up. If they won't tolerate a part of the exam that's absolutely necessary, then I step up to restraint/muzzle/drugs/whatever it takes. But you need to have some flexibility and work with your patient. As a caveat - sitting on the floor takes away your ability to move away quickly. Once you start to actually examine the patient, you need to change postures to something that allows you appropriate defensive mobility.
I got a thank you card a few months back from an owner whose dog I diagnosed with metastatic lungs and sent home for hospice care - she said in the card that after 14 years with their dog I was the first vet that sat down on the floor and took the time to talk to and pet her animal. It kinda sucks that that was her dog's veterinary experience for its life. I think a lot of vets storm into the room, talk to the owner for 30 seconds, then grab the animal and start pawing it over ..... we're all trained to 'establish relationship' with the owner, but it pays to spend just a little time putting the animal at ease, too.