I hike many rail-to-trail conversions on asphalt, sometimes gravel, thru the mountains, sharing the paved trail with roughly equal numbers of bikers. That solves many of your problems, while creating others. It would be financially advisable to research fees if any as tickets can be expensive. If you like bicycling in groups/crowds you enter the trail in a urban area, if you want to experience empty trails enter the trail from a rural farm county park.

I don't road bike but AFAIK the cheapest way to have a bike that weighs one pound less isn't hyper-expensive materials but diet based weight loss about a month ago. In the spirit of the old weight lifting joke, or maybe wisdom, that illegal steroid use might make you look like you have an extra year's lifting experience while causing significant medical problems, and in that context the medically safest replacement for illegal steroid use that gets you to the same place without any medical danger is simply to start lifting one year earlier, and a lot of money is made in the market convincing people that patience is not an alternative.