You can scroll up to post 74 and read the language in the Bill and draw your own conclusions, however, as I read it, it simply says a Med is not required to fly airplanes up to 6000lbs.
Since an aviator with a Sport Pilot Certificate is limited to Aircraft the meet the LSA requirement, I would conclude, No a SP wouldn't be legal to fly a Cherokee Six.
However, if this Bill or similar passes, a Sport Pilot going after a Private Pilot certificate shouldn't need a Medical.
Perversely, if you had access to a Night and Instrument equipped LSA, I think you could meet all of the aeronautical experience requirements in an LSA, however, the check ride for PP is required to be performed in a Certificated aircraft.