Edward did not want to name any women as his heirs. The original text of his Device for the Succession is very clearly naming the male heirs of his cousins. To quote from an earlier answer of mine,

Shortly before his own death, Edward decided to change this to keep out all of the women in line for the throne: Mary, Elizabeth, and the children of Henry's other sister, Mary, Duchess of Suffolk: Frances Grey and her children, all daughters, and Eleanor Clifford and her daughter. As he wrote up his "devise for the succession" initially, it ignored Frances and Eleanor and their daughters, but allowed them to transmit the right to the throne to their "heirs males". However, it was clear that there was no time for any of these women to get pregnant before he died, so in the end he altered the wording to allow the throne to pass to "Lady Jane [Grey] and her heirs males". As we know, of course, Mary did not take that sitting down, rose up in rebellion, and took the crown that she and many (if not most) others saw as rightfully hers.

The original text was "to the L. Janes heirs masles", and he then crossed out the "s" in "Janes" and wrote "and her" quite small after it. What Edward wanted was for the Grey sisters' sons to inherit, and the sisters' daughters' sons if they didn't have any sons themselves: however, he went into a decline and died much earlier than, uh, intended, so changes had to be made.

He also did not include either Mary or Elizabeth because at the time, they were both considered illegitimate - their mothers' marriages with Henry had been annulled (i.e. nullified), and while it had been the norm for annulled marriages not to retroactively illegitimate children born to them, Henry had done so in order to fully put those marriages in the past and make ti clear that they were not real. However, Henry had also restored them to the succession ... which was seen as highly irregular at the time, making the king's personal decisions more important than the common law. So from Edward's and his partisans' points of view, continuing to skip his half-sisters was absolutely justified and a much better decision than including them or just relegitimizing Elizabeth. It's hard to imagine being in that headspace because from a modern perspective, Elizabeth is THE queen, but she was simply a royal bastard for quite a while.

Historically, yes, the Duke of Northumberland has been blamed for putting Jane on the throne. I don't normally boost Eric Ives, but he makes a very persuasive case in Lady Jane Grey: A Tudor Mystery for Northumberland having been used as a fall guy by the rest of the administration, and a traditional Scheming Manipulator of a Good Young King for the history books, particularly important since Edward was a Protestant and traditional English history is quite biased toward Protestants, to be brief. I would strongly recommend that book for a discussion of the subject, and Edward's succession in general.