Apparently the author has not realized we have changed from a dispensation of law to a dispensation of grace. Things truly changed between the Old and New Testament, and the concept of priesthood is one of them. The Old Testament priest had to continually offer sacrifices for the sins of the people, because animal sacrifices were insuficient. In the 8th chapter of Hebrews, Paul calls the old covenant "obsolete, old, ready to vanish away". Later on, in the 10th chapter, he says that "we have been sanctified through the offering of the body of Jesus Christ once for all". What part of "once for all" is so hard to understand? We have a High Priest that "after He had offered one sacrifice for sins forever, sat down at the right hand of God" (Heb. 10:12). This is a new covenant, different from the old one, and therefore there's no need for priests, sacrifices...
The whole point why we don’t need sacrifices today is that the OT sacrifices were a foreshadowing of Christ, who still had yet to come, to be the sacrifice for ALL. Those animal sacrifices were based on the equity of Christ’s then-future sacrifice for mankind.
As Christ paid once and for all, we no longer need animal sacrifice, THE ONE REAL SACRIFICE those animal sacrifices represented, has been paid. No need anymore to unecessarily sacrifice animals.