Why are there many pilots who don't say "mayday, mayday, mayday" to declare an emergency?
I'm a controller, not a pilot, so I can only speak from my own perspective.
What we are taught in ATC school is that many pilots are reluctant to use the word mayday because they feel it might escalate a situation unnecessarily and potentially create a lot of paperwork. I guess, mentally, it seems like calling mayday is a significant, irreversible step which can be daunting to take. They might be afraid, assuming the situation goes well, that they have to explain why they decided to escalate the situation, get emergency services involved etc.
That's not really a resonable fear, though. In fact, as a controller, I would much prefer a pilot to call mayday as opposed to trying to make the situation seem less critical than it is. In an emergency, clear communication is absolutely essential. I've had several situations where pilots have had &quot;minor technical issues&quot; or an engine that is &quot;a bit jumpy&quot; or something like that, and it puts me in a really uncomfortable situation because I can't really tell how serious it is. And if I'm in doubt, I will just call the emergency services anyway. On the other hand, the one time I actually had someone call mayday, within 30 seconds, the firetrucks were on the move, all the runways were cleared and the aircraft in question was cleared to land. It's much easier to help you if you are clear about what you need!

Pilots believing themselves to be facing an emergency situation should declare an emergency as soon as possible and cancel it later if the situation allows.
The correct method of communicating this information to ATC is by using the prefix “MAYDAY, MAYDAY, MAYDAY” or “PAN PAN, PAN PAN, PAN PAN” as appropriate. This procedure, which is an international standard, is the single most effective means of alerting the controller to the need to give priority to the message that will follow.

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In the end, it is always up to the pilot(s) to decide what is the right thing to do. Every emergency situation is unique, and there isn't (and can't be) rules describing what exactly must be done in every single situation.