I, on the other hand, believe that the NT authors employ the OT to prove their point.
That's the problem with dispensational Zionist as yourself. You think the proper way to understand is by using that which was of Old to prove the New Covenant. You have it exactly backwards. It is the New that helps us to more fully understand that which was of Old. Of Old there was only one people of God, but that changed under the New Covenant. The people of God are not limited to one race of people but are also among the Gentile nations. And Gentiles as well as Jews are the strangers dispersed according to the providence of God. If the dispersion was of the Jews only, that is what would have been written. It makes ZERO sense for a Jew to write of Jews as 'strangers'. However, it makes perfect sense to call Christians without mention of race foreigners and strangers and pilgrims upon this earth!
Strong's Greek Dictionary
3927. παρεπίδημος parepidemos (parepídēmos)
Search for G3927 in KJVSL; in KJV.
παρεπίδημος parepídēmos, par-ep-id'-ay-mos
from G3844 and the base of G1927; an alien alongside, i.e. a resident foreigner:—pilgrim, stranger
Abraham is listed among all those of Old who died in faith. He lived and died before any were called Jews but he too was a stranger and pilgrim on the earth, among Gentile nations in unbelief. That because people of faith are always strangers and pilgrims here on this earth.
Hebrews 11:8 (KJV) By faith Abraham, when he was called to go out into a place which he should after receive for an inheritance, obeyed; and he went out, not knowing whither he went.
Hebrews 11:13 (KJV) These all died in faith, not having received the promises, but having seen them afar off, and were persuaded of them, and embraced them, and confessed that they were strangers and pilgrims on the earth.