Let's break this down logically. To say
A makes B cover C with D
you would say
In the first sentence mentioned in the question, not all of ABCD are immediately evident, but if you follow the link, you'll see that this is about what a teacher should instruct students to do, i.e.
The teacher makes the students cover the leaves with foil.
No problems using 被せさせる here.
The second example is trickier. We already have 絞首刑になる人に ending in に, but this could be either B or C. Also, A is not explicit. So either:
Someone makes people who get hanged cover somebody's eyes with blindfolds
Someone makes someone cover the eyes of people who get hanged with blindfolds
The first one hardly makes any sense, and in the second one, there doesn't seem to be any reason for the extra level of indirection.
Surely what is meant is one of the following:
They had the blindfolds that you cover the eyes of people who get hanged with.
They had the blindfolds that people who get hanged wear/put on.
They had the blindfolds that you make people who get hanged cover their eyes with.
So bottom line:
The first usage is correct, the second one is wrong.