From the realist POV the "person" is only the organization of neurons and their functions; none of the rest of the body really matters; and nearly every part have been severed by amputation or replaced by machinery or transplant without any alteration of "the person". (Other than normal personality changes due to persistently changed circumstances). Stephen Hawking has been a brain almost entirely disconnected from his body for most of his life; but is still a functioning human being. We transplant hearts, livers, kidneys, lungs, hands and more; even losing your heart doesn't change your memories, loves, politics or other inclinations.
We are just meat machines: Sometimes hateful, mean and cruel, but most of us are loving, caring, crying, romantic, joyful meat machines. This is excellent news for any meat machine that desires immortality.
Because our 100 billion neurons and their trillion connections are just biological entities rather like plants with intertwined roots. There is no magical component impossible to replicated.
Not only that, but the replication does not have to be perfect in order to preserve personality. Being of retirement age myself, I have seen Alzheimer's up close a few times among my parents and other older kin. Although it eventually does take the "person", a large amount of function can be lost while seemingly retaining the full "personality" characteristics of "who they are". My aunt had an unreasonably ridiculous love of dogs; and for years, even when she couldn't remember her own kids, they could get a reliably happy and joyful response from her simply by showing up to visit her with their combined pack of dogs.
Of course we do see radical personality changes when the brain is severely injured, by bullets or blunt trauma, cancer, other disease or surgery.
But the Alzheimer's patients are still good news for realists: It means the intertwining of neurons must have a very high redundancy, making the personality resilient to rather extensive damage. Of course we'd expect that to evolve; but in practical terms, it means replication can have a reasonable margin of error, in excess of 1%, and still reproduce the "person".
Presumably, some future technology (centuries away) really could freeze the brain, preserving the neurons and synapses. Then, in a destructive fashion, pick the brain apart and map every necessary component of it: Neurons, axons, synapses, glia, and anything else we have (by then) discovered is necessary.
The point is NOT to simulate this inside a computer, just to record the organization and network that is (in a realist POV) the personality, memories and emotions of Aunt Ariel at 50 years of age.
That information can be used grow a copy of her brain, minus any disease, with brand new "equipment". Something like with a 3-D printer, but over months, guiding the growth of the individual cells into the order specified by the "original" brain. Both recording and reproduction can have some small margin of error.
Also presumably in these centuries the biological control of cells needed to reconnect this brain to a new (brainless) body will all have been solved; and Aunt Ariel will awaken reborn.
From a realist POV that really will be Aunt Ariel; because all that counts and makes her, "HER", is that incredibly complex pattern of connections in her brain; which has plenty of redundancy and resilience to any damage caused by reproduction. So think of it as Aunt Ariel after being knocked unconscious in a minor car accident that caused no other injuries, and cured all her brain and body diseases in the bargain, and gave her a healthy 24 year old body.
Barring the collapse of civilization and technology; this could continue indefinitely, and would be effective immortality.
Perhaps with enough technology, the recording of every molecule and connection in the brain could be both non-invasive and diagnostic: so one could have an archive of themselves every day, like taking a shower, and computers could identify the presence of any disease or cancer the day it starts.
I would note we do not have to understand the brain in order to accurately reproduce it, just like a camera does not need to understand any meaning of the images it captures. Just like if I could replicate all the parts of a Swiss watch, in both material and form to the microscopic level, I could reassemble a copy of the watch that works like the original (otherwise I did not replicate all the parts correctly). There is no "soul" in the Swiss watch that makes it work; it does what it does as a consequence of mundane physics. The same holds for a computer. From the realist POV the same holds for animals and people; in principle they are a pattern of atoms and molecules that can be known and replicated.