It’s true that the author never establishes this, but he doesn’t need to. He’s only addressing what would happen if Mary sends in an application from Greendale. Whether she actually does apply is irrelevant.

It doesn’t matter how long it takes express mail to reach Gillespie City from Greendale. The author is only addressing what would happen if Mary sends her application by regular mail from Greendale.

What if the minimum amount of time it takes regular mail from Greendale to reach Gillespie City is five days? In that case, Mary might not need to send in her application ten days before the due date in order to be considered.

The author doesn’t assume that Mary will satisfy all the requirements if her application is received on time. She might mail her application on time and still not be considered for the grant. The author just argues that if she is considered, she must mail her application on time.

The author only addresses Mary sending her application by regular mail; express mail is irrelevant. Also, even if she can’t be sure that it will arrive by regular mail, this doesn't impact the conclusion that if it is considered, she must send it ten days before the due date.