I was about to say what you just wrote, tires are not all the same. Some will affect the bike's handling more than others.Those tires are part of the problem [IMO] I'm on PR 4 GT's currently and will soon be moving to PR 5 GTs. They're specifically designed for heavier bikes like the RT. Over the years, I've run tires because friends have suggested I try them.
I'm done playing the tire game because I've learned to stick with the performance - not necessarily the longest lasting tires - and I honestly couldn't care less about what I pay for the tires. I ride fairly aggressively and I ride slightly longer distances than the average rider.
Road handling is most important to me. Low end tires like the Shinko's simply don't offer the same performance as higher end tires.
Replace them with higher end tires and you change the bike significantly.
Michelin Pilot Road 5 GT Review