The version used is KJV
Ch8 and 9, which sets up the context for 10. In those chapters - 8:7 and 13, 9:1 and 18: Covenant was italicized in KJV and NASB, and in ESV it was part of the flow of text. Looking at TR covenant is not there
From context the passing away and replacement is the priesthood, therefore its the administration of the commandments that is talked of.
Covenant - terms in commandments, parties in Yah and Israel, penalties for violating commandments, restitutions for the violations, who can administer, where they can administer and when can they administer. How this changed with our now High Priest Yeshua
Once you have this, imagine youself to be a new believer youth in 64AD in a family on new believers. We were convicted by the second hand accounts of our Messiah and have given our life to Christ. We also are going to the temple to offer sacrifices. We have helped those in need (10:32-36), been ridiculed for faith, therefore Paul is admonishing that we should not fall away as we will see the physical temple going away. Our faith has to be in Yah and His Son, which is why the heavenly tabernacle and the Melchizadek priesthood was expounded upon. This is also why the wilderness journey was used to illustrate in Ch 4.
This then setsup the faith in Ch11, as we will have a crisis since we might have put our faith in a physical teample and the priesthood of men and the sacrifices.