But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.
It's the same reason why Obama couldn't be VP. He's ineligible because he's served two terms already. If you can't be President, you can't be Vice-President. And that makes sense, since the VP could become President at any time if something goes sideways.
I'm not a Constitutional studies guy, so I don't really know.
As far as I do know, the only requirement to be Speaker of the House is for a majority of the House to vote for you, and the only way the Speaker could become President is if something happens to the President and the VP simultaneously. I suspect what would happen in that case is it would skip over the Speaker and go to the next in the line of succession who is eligible.
A key duty of the VP is to become President if something happens to the President. It may not be a requirement, but it would cause the Speaker of the House, who might be from another party, to be the one who steps in.