The “keep a calm voice at all times” training definitely starts with the very first flight lesson. Aside from all of the above reasons - clarity of radio transmission, keeping the pilot calm, tradition, etc - a calm-sounding pilot helps keep the passengers calm. This is important. Imagine handling an emergency and trying to stave off imminent death while the people around you are having hysterics, weeping, puking, screaming, and so forth… very distracting.
I was on one commercial flight where the pilot’s voice did betray anxiety - and it was at that point that people panicked. Would have been much better if he had not said anything, really. Between smacking into something on take-off, the flaming engine, and other manifestations of Something Is Wrong Here we already knew the situation was serious, but up until the point the pilot used his I’m-about-to-piss-my-pants voice we were all sitting quietly awaiting our fate.
A calm voice really does help the guy next to you. A couple weeks ago I was flying home in an airplane with a dead alternator, a friend about halfway through his flight training along to practice his navigation and radio skills. Well, nothing electrical on the plane was working - no radios, no lights (but we were daytime, so not a problem), no navigation equipment - just the “how fast” and “how high” and “which direction” instruments working. Well, my friend is going over the “dead alternator” checklist while I keep flying and he says
“Hey, the electric fuel pump isn’t working!”
I say “That’s OK - we still have the engine-driven one.”
“But” - sounds of fear and anxiety, seat fidgeting, and other signs of potential Freaked Out Passenger " - what if that one quits working?"
“In that case” I say in a very clam, very matter of fact voice “Tighten your seatbelt, we’ll be landing about 5 minutes later.”
“Oh” he said, calming down “I guess so.”
Actually, we got home safe and sound with no further problems.