This statement is to try to clarify that the institution does not have a duty or is not to be held accountable for incomplete applications regardless of how an applicant may interpret the situation.
Students/applicants sometimes will blame others where it is not appropriate or enforceable. This statement is intended to make it clear that it is NOT the responsibility of the institution offering an application to ensure that incomplete applications are completed. That duty is with each applicant. Without this statement, it is possible an applicant could sue or otherwise think they can rightfully blame a third party for failure of duty in the process, especially if other statements or circumstances imply the institution or a third party is offering or has such duty.
For example, a student may assume that since their professor is an alumnus of an institution that they are applying to, that relationship as an alumnus creates a "duty" for the professor to provide a recommendation timely. Then if the professor fails to provide it, the student may try to place fault with the institution or professor because of this assumption, which may seem reasonable but it is not.
Another example is a system that sends a request for a recommendation directly to a professor based on an action by the student. A student may reasonably assume that once they have made the REQUEST for an item to complete an application, they have fulfilled their duty to complete that portion of the application. This also could seem reasonable but it is not.
Conversely, a request by a student to a professor to provide a letter should not be considered a contract based on the request, the application itself or a system that helps a professor submit such a letter. For example, a professor that is provided a login to complete a portion of an application on behalf of a student should not fear that a student could sue them for failing to provide the letter. A login or even receipt of that request might be seen by a student as a contract or other duty created by the system, and not one that exists solely based on their relationship with the professor.
In other words, "responsibility" in this context does not mean "control" but "duty" as in:
"it is your DUTY to make sure [someone with higher rank, like your advisor/supervisor/mentor] to submit/upload [something like a letter/approval] by deadline XXX" in order for your application to be considered complete.
It attempts to make clear that this duty does not lie elsewhere, as your question suggests. While you may believe such a request creates a duty, it only does so as a social agreement. An application, on the other hand, is usually a legal contract and as such there are obligationsand duties that are often legally binding and enforceable.
You may feel like "responsibility" means "control" but it means "to have a duty" or to be the party that is accountable. Most often, the only other party in an application is the institution. A person providing content/letters on behalf of a student for an application is almost certainly not a party to the application.