How The Master Cheats Death
Just to get it out of the way, no, I don't really think it matters exactly how The Master escapes certain doom between appearances. That's just their character. they always comes back. Like that case of long covid you've been fighting for the last 4 years, every time you think it's gone it rears its head again and makes your life a living hell for a little bit.
But just because I like making headcanon's, I felt like I would think this out a little bit.
Also note: I will be mostly focusing on the modern era simply because my knowledge of Classic Who is very limited. Also, most of their death dodging escapades can be explained by black magic and overly convoluted body swapping schemes.
The first major almost definitive death of The Master comes in the 1996 movie. At the end of the movie, Eric Roberts' Master is pushed into the eye of the Tardis and therefore the Time-Stream where he was likely shred to pieces.
The Master's then return in the modern series is actually somewhat explained in The Sound of Drums. During the phonecall between himself and the 10th Doctor he alludes to the Timelords resurrecting him to be used as a weapon during the Timewar. This also explains how the Master regenerates. When we first met The Master during the Third Doctor's era, he was already on his last regeneration. Hence his need to find unnatural ways to artificially extend his life rather than simply regenerate. Even the Eric Roberts version of the character is meant to be the same regeneration that Roger Delgato first played in 1971. Simply wearing Bruce the EMT as a meatsuit. Derek Jacobi plays the first fully new incarnation of the character we had ever seen on screen. Something we don't often actually talk about.
Expanded media with tenuous canon authority explore The Master's time in the timewar more expansively, but it shows that the Master had several incarnations during the timewar. At least 3. Which proves that the timelords resurrected them with a brand-new regeneration cycle. Eventually the timewar even scared them off. Finding some means of escape before Gallifrey entered the timelock. We know that the Master wasn't there during the final hours of the timewar. As the Saxon Master was unaware of how savage and insane they had become in The End of Time.
He plops himself as near to the end of the universe as possible in hopes it was the last place the Timewar would find itself. What is the point in war if there is nothing to claim or conquer but dusty dead rocks and dying stars?
He chameloen-arches himself. He's stuck with his current face but can at least reset the biological clock, and enters the society of human survivors as a child. The events of Utopia occur and he regenerates into Harold Saxon. Sound of Drums and Last of the Timelords happen and despite having at least a few more regenerations left, instead chooses to just die. (Which is something timelords can do apparently.. if they are stubborn enough)
But The Master is back to old tricks. That year he spent building starlink....er.. I mean ArcAngel Network and building power was apparently also spent building a cult of personality that then uses magic to resurrect The Master once more. Albeit slightly poorly. By the end of The End of Time, the broken Simm Doctor is trapped once more in the Timewar.
This presents 2 situations. Either A) The events of the 50th occur and Gallifrey is never destroyed and trapped within a pocket universe which The Master somehow escapes. Or B) The Master didn't stay for very long. Just long enough to fix himself with Gallifreyan tech and then steal a Tardis and escape the timelock.
B being my personal interpretation. Escaping Gallifrey's timelock did serious damage to his Tardis. He essentially had to immediately emergency land on the Mondasian colony ship. He does as 12 states. Finds a population, subjugated them, and then they had an uprising and he went into hiding. Seeing where the colony ship was designed to go, he just couldn't help himself and from the background created a cyber-conversion movement by capitalizing on the poor living conditions of those living at the bottom of the ship (conditions he partially created when he was in charge) World Enough and Time happens and Missy betrays him.
Here is where things get complicated. See, after Missy stabs Saxon Master, he retaliates by shooting Missy with his laser screwdriver which also disabled her ability to regenerate
Meaning, this should be the definitive end to The Master. Afterall, if anyone was ever going to actually kill The Master, it had to be themself. Except, this isn't the final incarnation we see. That still leaves Sacha Dhawan's Master.
I know for a fact I am not the first one to suggest this, but Sacha's Master might come before Missy.
When the Saxon Master regenerates, he's going to be more pissed than he has ever been before. Here we have a raging narcissist who has just been betrayed by himself. The literal only person he loves. This is what ultimately sends The Master down a spiral of violent instability (at least more than usual) and self-loathing. A new feature for the character.
The Master has always been an egoist and a narcissist. They really only care for themself, and view themself as better than literally everyone else. The Master very rarely shows any modicum of respect for anyone else. With only The Doctor and handful of other Timelords ever being seen as an equal to them. Self-loathing does not fit that profile. But by the end of Dhawan's portrayal, that is what is left. A Master driven by jealousy and envy. So eager to take The Doctor's face and leave his own behind. Not out of any need to escape death like with the 1996 movie. There is no actual need to steal the Doctor's face. He simply wants to. Because he no longer likes his own.
It stand to reason that a narcissist can only exist at the two extremes. They can only be absolutely perfect or absolutely flawed. Nothing in between. A narcissist who falls out of love with themselves would then completely hate themself.
Another key factor is The Master's hatred. The Master has always had a rivalry with The Doctor but for most incarnations it was mostly a game. A battle of witts. The Doctor is the only other being in the universe that The Master deems as a true equal, and therefore feels the compulsion to best them. They've always been in a sometimes-deadly intergalactic game of Uncle and nothing more. they just want to defeat The Doctor to prove they are the best. It's never truly personal. If it wasn't The Doctor it would be someone else.
Except for O. Sacha's incarnation has a genuine hatred of and vendetta against The Doctor. It isn't a game anymore. It's revenge. He's not torturing 13 to get her to give up so he can win their cat-and-mouse game. He wants to destroy her. Break her.
Here's how it goes. Saxon Master regenerates into Dhawan Master. Extremely emotionally unstable. It's Sacha's Doctor that finds Gallifrey in its little pocket universe and springs it loose. Hangs around for a bit (Long enough for the events of Hell Bent to take place) Does some digging and eventually destroys it. Like a toddler having a tantrum.
At first he simply wants to hurt 13. She is the co-conspirator in his pain. Afterall, it was The Doctor who Missy betrayed themself for. She is the Destroyer of everything The Master is. She must be punished for what she did to him. But that pesky self-loathing. It slowly grows from hatred, to envy, to jeolousy. Eventually, the worst fate The Master could imagine was being himself. And so he concocts a plan. To give The Doctor his face. The only thing that could truly free him was to become someone else. And so it goes with The Power of the Doctor. Dhawan's Doctor forces them to swap bodies. And not out of any survivalist need. Here he simply wants to. Both to spite The Doctor and to finally escape himself.
This, of course, doesn't work. The Doctor escapes with her body and The Master regenerates again. This time into Missy (supposedly, until a new incarnation shows up and must occur between them). Either way, when The Master finally reaches Missy, it becomes the straw that broke the camel's back. She can no longer deny or compartmentalize what happened or what will happen. When she looks in the mirror, she sees the part of herself that she hates most. The one that betrayed her... and the one that will die.
This is what forces The Master to re-evaluate herself. She becomes a new person, with a new name. Missy. This is a Master who recognizes that she has always been the architect of her own pain. That she is lonely, and that the only constant companion in her life is The Doctor. So she uses the very tools that caused her destruction to fix it. Once again using the Cybermen as part of her scheme. But instead of causing chaos, she uses them to show her appreciation for The Doctor (In her own twisted and morally/ethically dubious way). It's not death simply for the sake of death. In her mind, she brought the entire human race (The Doctor's most beloved "pets") back from the dead. A true gift to him. Completely unconcerned that the form in which the deceased have taken is a desecration to their memory and their loved ones. I mean, they are a standing army. You could invade the universe with them. "Surely that is better than some rotting corpse?"
At the end of the day, Missy still struggles to conceptualize the meaning of life and empathy for others beyond herself. Her psychology is so fragile and limited she only barely made room within it for The Doctor. And her universal solution to most problems (violence and subjugation) still inspires every action she does. What point does a person have if they can't be a pawn in your scheme?