What happened to riding with no hands? I remember, as a kid in the seventies we were able to ride blocks without ever touching the handlebars.
Today, I only see road-bikers riding without hands - and this is rarely.  I almost never see hybrid or mtb bikers go without hands.

What happened? Is it because I got older that I don't have the sense of balance anymore?  Or are bikes made differently in the sense that they're not self-balancing? Or is it the forward positioning??
Nothing happened or changed. It's just your observational bias coupled with different demographics preferring different bike capabilities.

Many children's bikes have relatively relaxed geometry, which makes them stable which has benefits for kids. Bikes built for adults usually aren't as stable, because stability and maneuverability are at odds with each other, and the adult riders want the maneuverability. 

Mountain bikers value fast handling over stability, and so do many road cyclists.

However, many triathlons or time trials are events where maneuverability is not important, so many of those frames are much more stable. Touring bikes are also in that group.

I have a bike built up from a nashbar touring frame, that is so stable I can nearly fall asleep while riding. I hop off that bike and hop onto an 80s Bianchi racing bike, and trying to ride no-handed at any speed less than 35 mph (55 km/h) is virtually an instant crash. My surly cross check is in the middle of that range.