Let's consult The US Constitution (<- That's a link) for an answer.
Section 7 - Revenue Bills, Legislative Process, Presidential VetoAll bills for raising Revenue shall originate in the House of Representatives ...
Now, if a bill for raising revenue was the same as a spending bill, the answer would be yes. But in fact, a bill for raising revenue is a tax bill, rather than an authorization or spending bill.
So, I conclude that the answer to the question you posed, "Do not spending bills have to originate in the House, per the constitution?" is a big, fat, resounding
Out of the 10 or so Senators who spoke today, all of them spoke at least a little bit on the bailout, and all of them were in favor. McConnell, Reid, Domenici, Kyl, Bond, Bennett, Alexander, Warner and Dodd. Dorgan spoke out, not in favor (check him out on Nov 4, 1999 on the same subject! (November 4, 1999).
The only procedural issue is one of timing. Any Senator can delay a new bill by objecting to proceed to its consideration. That was impossible with the bill the House just rejected, because the House used a procedural trick of putting the new bill in an old package.