Personally I find my fuel economy increases with new tyres, and the reason that is because I generally drive with performance in mind. New tyres have less grip so I generally drive slower until I'm happy with the grip levels, so that gives be an economy boost for a short period.
Different tire models have different load ratings at the same pressure. The tire pressure on the car provides a certain load rating for the OEM tire. A different tire with a different load rating, or different max pressure, can have a different rating at the pressure specified by the car. One that is lower than what the manufacturer designed to car for, and could be over loaded at the car's rated gross weight.
Insurance companies will find any excuse not to pay out on a claim. When I was younger, I had Italian stripes fitted to my Abarth 500, I was told I need to inform my insurance company because my car now draws the attention of theives more. With my Peugeot, I wanted to remap it, only an extra 30-40 bhp so she's running around the 300 bhp mark. My insurance quotes jumped by 100% just for adding that mod so I didn't bother. It's this kind of greed that encourages dishonesty amongst its clients, people will just stop declaring modifications...
Thanks for the insights regarding getting new tires. I was thinking of acquiring wider tires from 4WheelOnline to replace my current worn-out ones. I want to reduce the tire wearing as well. If possible, I want to replace my stock wheels too.