Are 10-year-old winter tires OK for summer use?

Unfortunately, not really. The concern with 10 year old tires is degradation of the rubber leading to increased risk of failure. This degradation happens regardless of how much you use them. Tires are generally manufactured to last about 5 to 7 years, usually the tread wears out before the rubber goes.

As you noticed the rubber has already changed, once you get them up to highway speed or otherwise get them hot there is a high risk of sudden tire failure. That softness you've lost is a key part of why we use air filled rubber tires, so they can flex and distribute force across the whole tire, especially at speed.

The ride will probably be pretty bad also, since tires won't flex the way they were intended anymore.

This is a pretty common topic of discussion in the RV community, so as an RV'er I see variations on this question a lot. RV's aren't usually driven much, so the tires need to be replaced long before the tread is worn out. It's a bit painful because the tires look great, but an RV is heavy and they are no longer safe.

In some special cases, like if you only use the car for local non-highway trips where a flat wouldn't be that big of a deal, it's less of a concern.

Don't judge by appearance, judge by feel. Even if they have low mileage and look good, they are not. But you can still get some mileage out of them if you go easy on them. Just be careful and check them frequently.

I think that's how Paul Walker died. The Porsche he was in had wheels that looked great/brand new, but they were in fact old, and couldn't handle the speeds/cornering he was doing.