This is probably an unpopular opinion, but in my view, traditional final exams tend to be a great equalizer. Continuous assessment is affected much more by personal circumstances.
For example, continuous assessment tends to force students to devote a constant amount of time throughout the course, which is fine for those who can afford to be full-time students, but disastrous for students who need to have a job to pay the bills. I have seen plenty of students like that getting frustrated because they lose points to this.
On the other hand, continuous assessment typically relies on assignments at home and these are deeply unequal and classist as well. The difficulty of these assignments depends on whether the student has someone more experienced to ask, whether they are paying for a private tutor, or whether they can plainly buy the assignment. I have known cases of students that basically bought every project and assignment that could be done at home, you know what made them sweat because it couldn't be bought? The final exam.
But almost no one cares about these things, continuous assessment is seen as the "progressive" thing to do because it's more modern and different from tradition, even if it's extremely classist and disproportionately harms working-class students.
On the other hand, it is true, as you point out, that the final exam model disadvantages students who have the bad luck of getting sick in that week. It's a real drawback, sadly. I still think it's better all things considered, for the reasons above.