I recently bought this old Honda Tact from a neighbor as a fixer-upper. Here she is:

As I understand it, this is a Japanese version of the Aero 50. It's almost identical in many of its components, but there are some differences.
One, is this:



As you can see, I haven't got an Autobystarter like most Aero 50's. Instead, i've got this rubber thing. Currently, the only major issue with my scooter is that it can't start cold. It takes about 15 minutes of patient kick-starting to slowly warm up the engine. Once it's warm, it runs beautifully.
Now since this doesn't have an autobystarter, is this rubber thing supposed to help cold starts? I've tried playing around with it, and noticed that if I block the air vents (seen in the 2nd closeup photo), it seems to start a bit easier when cold.
Anyways, have any of you seen this before? If so, any idea how exactly it's intended to work?
Any comments are appreciated, thanks!