Sure you can spin it on bike as well. Have to loosen axle-nut so there's no pressure. Also remove brake-caliper so pads don't rub. There's still some friction from bearing-seals and axle-seal, so won't be as free-spinning as balance stand.
Other thing that may at work here is fix-a-flat sprays. I've seen lots of wheels that had this crap sprayed in to seal minor leaks. Problem is most of it stays liquid inside tyre and heavy spot moves around. This causes crazy bouncing of wheel and vibrations to your hand, and it can't be fixed by balancing!
Purpose of sidewalls is to provide tension against air-pressure (think of a suspension-bridge rolled up into donut-shape). This tension is carried by the fabric cords in sidewall. Cord tension is reduced by weight of bike at contact-patch area on bottom (thus why tyre bulges out). Reduced tension there is carried by air-pressure and increases tension along all other cords not in contact area. Sidewall rubber itself doesn't take much load and its main purpose is to seal against air-pressure since fabric doesn't seal too well against air-pressure.
In cases of performance tyres, rubber layer is minimized so much, you can actually see fabric pattern showing through sidewalls. In extreme cases, there's no rubber at all on sidewall, just thin sealant infused into fabric itself (skinwall tyres on racing bicycles). Best tyres I've ever used was 100gm silk sew-up slicks on my velodrome bike. You can actually see light through sidewall of tyre! And tread had just enough rubber for 6-laps, just 2-km !!!
Yeah, those tyres are fine to use. Don't be too aggressive, however I was able to regularly scrape centre-stand and exhaust on my 2002 with those 15-year old tyres. Smooth is fast. Tyre is "bad" when it no longer grips. This is caused by thin rubber layer which does not have enough flex to transmit hysteresis forces into inner layers. With experience, you can tell when tyre-grip is going off by gradually increasing speeds until it slides. Worn tyres will slide at lower-speeds with different "feel". Cracked sidewalls are also bad because eventually cracks will spread to fabric layers and you may have rapid decompression. Although more likely, it'll manifest as slow-leaks that can last for years.