Does having a UK visa, of any sort, help improve my chances of applying for a Schengen?
Yes, having a current, valid UK visa is an absolute plus and lends credibility to your application.
It not a guarantee however, and you will still need a viable premise for visiting the Schengen zone; and you will need to submit a well-formed itinerary (along with evidence of your itinerary).
You wrote that your premise to drive through cities in a shared car without stopping. This is a bit unusual and you may want to explain why that kind of trip is interesting at this particular point in time. Based upon what you wrote its almost certain they will be looking for "Why now?". Also, include your budget for tolls and your bookings for the channel crossing.
As a general rule as your UK visa nears expiry its value for supporting a Schengen application rapidly diminishes. When its expiry is about three months or less, you will need a really strong premise and a solid itinerary. It's reasonable for them to curtail your Schengen so that its expiry coincides with the expiry of your UK visa.