Objection to Luke's genealogy stemming from Mary

On p 46-7 and 97-99 of EBE Dennis McKinsey waxes eloquently on the errors in the two genealogies of Jesus in Mat 1 and Luke 3. He objects to the argument that Matthew traces his ancestry through his legal father Joseph and Luke his ancestry through his actual mother.

1. Mary's name is not found in Luke's genealogy
2. There is no genealogical record of any woman in the bible
3. Joseph's name is mentioned in Luke's genealogy and Luke elsewhere (1:27, 2:4) shows that Joseph was descended from David, so it is fair to conclude that Luke's genealogy belongs to Joseph.
4. If Luke's genealogy were that of Mary she would be disqualified as because of Shealtiel and Zerubabbel the descendants of Jeconias who would never have a descendant on the throne of David (Jer 22:28-30).
5. As a woman Mary could never be qualified to be heir to the throne of David.
6. The Messiah would be a physical descendent of David (2 Sam 7:12-13, Psa 89:3-4, 132:11), but Mary seems be of the house of Levi since her cousin Elizabeth (Luke 1:36) was a daughter of Aaron (Luk 1:5)
7. According to 1 Chron 22:9-10 the Messiah had to be a descendent of the throne of Solomon. But Luke's genealogy is traced through Nathan another son of David.

Related difficulties:

The two genealogies of Jesus Mat 1, Luke 3:23 (12k)

The Messiah must be a physical descendant of David (6k)

See Glenn Miller's answer here, see CARM's answer here