"What is the greater symbolism and purpose of these death scenes?"
Those death scenes are - in my opinion, as there seems to be no official explanation (as repeatedly lamented or critiqued in threads concerning this topic) - a condensation of the information collected (intentionally as well as intuitively) by the assassin, presented in a compressed and abstracted form in a virtual "white room".
This could have been done for the sake of gameplay (either ours or Desmond's, or both), as letting the player collect all this information would probably have been a tedious and time-consuming task, both for us and the developers.
The virtual room is either a purely symbolic device for us as players, or, in-universe, a kind of 'hub' where the information is relayed to Desmond.
Another way these scenes can be explained, and one that's not inconsistent with the last, is that the Assassin's supernatural Eagle Vision is able to create a telepathic link with the victim. After all, the assassin's do have supernatural abilities, and this might be one of them.
These death scenes are generally not (to be) seen as confessions, which is, for one, fairly obvious, considering the directness of the lethality of the signature hidden blade with which many of the victims are killed.
Besides: why would templars start spilling their guts as their guts are being spilled? After all, many of these victims were not repentant, stressing the earnestness of their ways.
As mentioned in the comments, wiping the blood of the victim with a feather was a way for Altaïr, and Bayek before him, to proof the killing to the Brotherhood.
These mannerisms of the Order of Assassins have disappeared or changed throughout the centuries: the Frye twins use their handkerchiefs - most other assassins don't bother, or take trophies from their victims.
On page 6 of Altaïr's Codex we can read the following:
It's made me realize that our tactics, too, must change. It means an
end to our fortresses. To our penchant for spectacular displays of
public assassinations. We must weave our webs quietly. And we must do
so differently than we have in the past.
Though I ask my brothers now to abandon their rituals, I do not ask
that they abandon the creed. This is what makes us assassins. Not the
removal of a finger. Not a false promise of paradise. Not the
prohibition of poison. Our duty is to the people, not to custom. If we
must sneak, we shall sneak. If we must use poison, we shall use
poison. If our blades can be used without removing fingers, we shall
not demand they be taken.
And, after all (p. 4):
Nothing is true. Everything is permitted.
Ultimately, it doesn't matter how their ideologies are maintained.
"Why doesn't Jacob wipe the blood with the handkerchief?"
As for Jacob's abstaining when it comes to taking blood from Malcolm Millner, it is, like Wouter stated in the comments, unnecessary to prove this victim's death, as Jacob was misguided by Pearl Attaway who had ordered Millner's death, but was, unknown to Jacob at that time, involved with the Templars (source).