Another angle on this endless debate... (I think the OP is long gone and decided to change the tires, FWIW.)
I'm firmly in the "replace them" camp. One reason for this is mental: if you are wasting one iota of your brain's bandwidth thinking about your tires or if you have the slightest doubt in the quality of the rubber beneath you, it WILL negatively affect your riding and your safety. Spend the money a little sooner so you can concentrate 100% on riding. If you're really frugal, learn to change your own tires; you'll save a ton of money and you'll do a FAR better job overall than any shop.
Secondly, statements along the lines of "I don't ride aggressively, so I don't need to worry about slippery hardened old tires" are complete nonsense. In a post-apocalyptic wasteland with no drivers, animals, children, trees, dirt, etc. then perhaps you might get the chance to choose your environment and how much traction you're using 100% of the time. Or maybe you only ride gently on a gently curved road within the confines of a private fenced estate in Kansas or something.
Out here in the real world, you do not get that choice. Even the mildest milquetoast rider WILL frequently encounter the need to maneuver and brake hard and MUST be ready and able to do so.
You see the same flawed reasoning when discussing suspension upgrades, brake upgrades, skills improvement, etc. "I don't hoon around like some of you animals, so Suzuki's entirely inadequate springing doesn't bother me".