Priya, our anxious dog ended up flying in the hold because she was too big for the cabin, and I was terrified for weeks, so here's how it went, honestly.
We used a specialist pet relocation company for the paperwork and the airport part, because I didn't trust myself to get the timings right. They helped us choose a direct flight at a cool time of day, told us exactly how to prepare the crate, and handled the drop-off at a separate cargo area, which I would never have found on my own.
Leaving her was the worst part. I cried in the car park. But when we picked her up at the other end, she was calm, a little confused, and very happy to see us. The crate training we'd done for a month beforehand made all the difference, I'm sure of it. She slept for most of the next day and then behaved as if nothing had happened.
So my honest answer: the fear was much bigger than the reality. Prepare the crate well, choose the flight carefully, and let the professionals handle the part you can't.