Lewis' resolution is based on the distinction that one 'could' (have the physical capacity to) kill his or her grandfather but, ultimately 'would not' be able to do so (without creating a paradox.)
We could consider the Novikov self-consistency principle, which assumes that there is a single timeline (thereby avoiding the many-worlds theory,) in order to argue that one could both have the ability to and realize the eventuality of killing one's own grandfather.
It may seem counter-intuitive since this is the principle that is usually referenced to disprove the outcome of killing one's grandfather (e.g. one would die, or be arrested or the like, before being able to kill his or her grandfather.) However, the universe may still conserve itself within its CTCs by allowing certain situational circumstances. Let's use Lewis' own "Tim the time traveler" for the examples.
For the sake of argument, let's say Tim travels backward in time and kills his grandfather. Despite this, he still continues to exist...
Circumstance A: Tim's grandfather died at a relatively early age.
(advanced medicine theory) Before Tim is able to complete his diabolical plot, his grandfather visited and donated to a sperm-bank. Later, after Tim kills his grandfather, Tim's grandmother receives his grandfather's sperm.
(secret technology theory) Tim's grandfather donates his body to science. After his death, a government entity uses the grandfather's DNA to successfully clone an exact copy of the dead man. The "resurrected" grandfather goes on to conceive one of Tim's parents at a later time.
Circumstance B: Tim kills the wrong grandfather.
(case of mistaken identity theory) Turns out that the man that Tim knows as his grandfather is not his biological grandfather... Tim's grandmother was unfaithful to his grandfather, thereby making a cuckold of him. (This may be a bit of a loop-hole that is unsatisfying to the principle of the grandfather paradox but, this sort of thing happens in society. The chance of this mistaken identity increases statistically if one considers that perhaps Tim's mother was also unfaithful to his father instead of, or even in addition to, considering the grandmother's adultery.)
(the twin theory) Tim kills his grandfather's identical twin. Since losing his brother was so difficult for Tim's grandfather to live with, he never spoke about his brother to his wife or children or grandchildren.
GAME OF THEORIES:
(Weiner is coming theory - from circumstance A) Tim's grandma is a coroner and a bit of a sicko. After Tim kills his grandfather, the grandfather's body is delivered to the grandmother's morgue. Under the right circumstances, a dead body can still be able to achieve an erection and ejaculate...
(Lannister children theory - from circumstance B) Let's assume, for the sake of argument (and because we share a name,) that Tim is not a sicko himself. And yet, killing his grandfather had such an impact on him that he subsequently went out to get totally wasted in order to forget his troubles... Unfortunately, he chooses to go to the same bar as his grandmother that night. And in their inebriation, they both make a terrible mistake. OR, without considering the consequences, Tim himself decides to visit and donate to a sperm-bank from which his grandmother later receives his sperm. In either case, the man that Tim's grandmother subsequently marries, and Tim comes to know (and later kill) as his grandfather, is not his biological grandfather (This would create a causal loop which could be considered to be a paradox. But, this then takes us back to Lewis' arguments in which he says that a causal loop may be inexplicable [like the Big Bang] but, not necessarily impossible.) (1)
Vranas didn't seem to like consistency paradoxes either but, he did consider them "worthy of serious scrutiny." (2)
However after analyzing a solution, by Fernando Echeverria and Gunnar Klinkhammer, to the Polchinski's paradox scenario of the Novikov self-consistency principle, Kip Thorne and Robert Forward:
illustrated that for certain initial trajectories of the billiard
ball, there could actually be an infinite number of self-consistent
Echeverria, Klinkhammer, and Thorne published a paper discussing these
results in 1991; in addition, they reported that they had tried to see
if they could find any initial conditions for the billiard ball for
which there were no self-consistent extensions, but were unable to do
so. Thus, it is plausible that there exist self-consistent extensions
for every possible initial trajectory, although this has not been
If that is the case, then any possibility of self-consistency should be examined until it can disproved.
1 - https://usdintrophil.files.wordpress.com/2019/03/lewis-time-travel.pdf
2 - http://fitelson.org/125/vranas.pdf
3 - [https://en.wikipedia.org/wiki/Black_Holes_and_Time_Warps]
4 - https://authors.library.caltech.edu/6469/