I think people want TItle II partially as retaliation for the ISPs being as terrible as they are. Price/MBit hasn't gone down much(any) for me over the past 10 years. If Google can have 0 infrastructure in a town and wire it up at gigabit speeds, why is it so hard for the ISPs to improve? Its because they don't care have 0 incentive.
Maybe we set a moving limit. Internet connection under 50Mbit /down or < 10 Megabit up is classified as title II. Maybe then they'll have incentive to innovate. 2 years down the line its 50/15... I don't know.
I think people want Title II because they support the general direction of the 2010 FCC regulations, and recognize that those are very much common-carrier style regulations and that there is significant doubt that they can both pass legal muster under Title I and retain their substance
What really needs to happen IMO is "none of the above" - Congress needs to get its act together and pass a revised Telecommunications Act that reflects reality, so that the FCC has more flexibility to regulate beyond choosing a regulatory box to fit ISPs in ("information service" and the weak regulation that entails, vs. "telecommunications service" and the common carrier overhead that entails).
That's a more proper solution, but it is pretty clear that this Congress isn't able to do anything of the sort, and IMO we can't wait for another one to come along.
The route via 706 was just struck down in court. Hence the next routes available are a) allow fast lanes or b) do not allow fast lanes via title II.
edit: I think that the bill you are referring to is Leahy's which would only give political support and not really give it new authority.