I see. Every time the bible is saying something crazy, we must be misinterpreting the obvious interpretation.
"Without father, without mother" doesn't make sense applied to "the priesthood", because of course no one would assume that a title or role would have parents. "Having neither beginning of days nor end of life" doesn't describe a lineage, because lineages clearly do have a beginning (and allegedly will have an end). Hebrews 7:1-4 constantly refers to a single man by the name of Melchizedek who had several roles (king of Salem, priest of the Most High God) and who was "like unto the Son of God", assuming that the son of God was an immortal right from the start who wasn't born and who never died. This is the obvious interpretation, and if the author meant it differently then he wrote very poorly. But if you have an argument against this clear meaning, then you really ought to cite it with something more than assertions.
Furthermore, there is no "priest hood" of Jesus. Nothing that Jesus did filled the role of a priest, and Jesus didn't have a lineage to continue this role that he never took on. In fact, if you believe that Jesus died as a sacrifice, then Pontius Pilate fulfilled the role of a priest and Jesus was just the lamb -- not the person who killed the lamb.