Many industrial robots perform tasks that are ill-suited either to hum

"that" is not really "modifying" anything. It's like this: He was so weak that he could not walk.

Clearly the conjunction used here is “Either X or Y”. Everything outside this “either…or” structure is “common” to both X and Y.

Option A: Portion outside the “either…or” structure: Many industrial robots perform tasks that are ill-suited

X: to human hands and eyes
Y: are so onerous or strenuous that people do not want to do them.
But the common portion does not go well with Y: Many industrial robots perform tasks that are ill-suited are so onerous or strenuous that people do not want to do them - Seems illogical. Hence, A is incorrect.
B and D end in “so”. This seems to suggest that the tasks are so onerous or strenuous “so that” people do not want to do them. The presence of “so that” seems to imply that the tasks are “intentionally” onerous or strenuous in order that people do not want to do them! This is clearly absurd.

Option C: Portion outside the “either…or” structure: Many industrial robots perform tasks that

X: ill-suited to human hands and eyes
Y: are so onerous or strenuous that people do not want to do them
But the common portion does not go well with X: Many industrial robots perform tasks that ill-suited to human hands and eyes - Seems illogical again (it should have been: “..that ARE ill-suited…”). Hence, C is incorrect.

Option E: Portion outside the “either…or” structure: Many industrial robots perform tasks that

X: are ill-suited to human hands and eyes
Y: are so onerous or strenuous that people do not want to do them

Now, the common portion goes well with both X and Y:

Common portion with X: Many industrial robots perform tasks that are ill-suited to human hands and eyes
Common portion with Y: Many industrial robots perform tasks that are so onerous or strenuous that people do not want to do them