tl;dr a partial/localized immune response just in the nose; and to confirm the vaccine worked you want to be sure
It seems "the basic issue is that immunity response in the mucus lining (i.e. nose, lung, airway surfaces) can occur independently of response in the bloodstream".
Which I guess means a false positive for post-vaccination verification. (The sentence you have copied is shortly after this: "The key problem is how to check that the vaccine worked.")
If your question is why it's possible to have a partial immune response, then it's because the immune system was not activated fully. The immune response might only consist of "sIgA antibodies in nasalwash and saliva", sIgA is secreted IgA, and IgA "is an antibody that plays a crucial role in the immune function of mucous membranes".
So if mucous membrane successfully "handles" the vaccine, then -- layman speculation -- it's possible nothing [not enough] gets into the bloodstream/lymph.
I took what the author wrote to indicate that there was no available test to do what he'd wanted to do. I really wish people would drop the whole "because reasons" thing, especially when writing something meant to be read by many other people.
I'd really like to know what those reasons are. It seems like a pretty important detail.