In the analogy of Jesus and sheep, Jesus not only compares himself to a door, but he also compares himself to a shepherd, which is compared to a hireling.
According to The Free Dictionary, a "hireling" is "one who solely works for compensation". In particular, in Jesus's analogy, Jesus talks about how the hireling does not care for the sheep (10:13) -- likely because of a greater focus on rewards. Additionally, Jesus talks about how the hireling flees upon seeing a wolf (10:12), and such action is likely linked to a lack of care for the sheep. In summary, because the hireling only cares about oneself, the hireling is not meant to care for the sheep -- and the hireling represents the leaders who only care about themselves rather than the people they lead.
In contrast to a hireling, a shepherd takes care of the sheep, and Jesus establishes himself as a shepherd, who takes care of the sheep (AKA people who are receptive of God). In particular, Jesus talks about how the shepherd "gives His life" and "la[ys] down [His] life for the sheep" (10:11, 15) -- which references Jesus's attitude on service (especially with what he would later do on the cross). Jesus also talks about how he knows the Father and vice versa (10:15), something further explained in past sections like in John 5. Not only that, Jesus talks about how the sheep know Jesus and vice versa (10:14). Everyone who has accepted Jesus has known Jesus to some extent.
Additionally, Jesus talks about how he would bring other sheep not in the current fold so there can be one flock (10:15). This is a reference to how while there are many people of God all over the world (especially the Gentiles who later accepted Jesus), Jesus brings them all together as one group under Jesus. Jesus is not just a great leader, but a great servant and unifier for our behalf.