I think you may be taking it to an extreme, it was inevitable that powerful phones that could play game would steal some of the handheld market but I wouldn't go so far as to say it doomed all handhelds. Just as it was inevitable that netflix would steal money from cable providers, but then it became the job of cable providers to adapt (something they are doing poorly).
However, just because many of technologies have died because they failed to evolve with the times doesn't mean handheld gaming devices will, in fact Sony and Nintendo are both very aware of the changes and have each developed new strategies.
*Apple has designed a phone that can play games. Problem is the games available are nowhere near as powerful or well developed as the competition, and will have a very hard time luring real gamers away from their current devices.
*Sony move the same direction but from the opposite starting point and is creating a gaming device that can do the same functions as a smart phone. This allows gamers to still play their favorite games while being able to get rid of their smart phone, however, it will turn away people who want the their device to be for more serious applications.
*Nintendo on the other hand has taken what I think is a very smart move, while Sony and Apple head in the same direction where one inevitably has to win, Nintendo has decided to distinguish their device and go their own direction. While many would argue Nintendo is doing a poor job currently, I would like to remind people how Nintendo has come from behind and turned a silly childish idea into a huge success. Just because they released one flop doesn't mean they have met their end, many didn't really like the GB Advanced but then they release the GB Advanced SP and sales shot through the roof.
The time may come where technology advances so far that there is no reason why our handheld device doesn't have a phone in it but that time isn't now. Apple may want to claim they are winning but Nintendo would state that they aren't really competing in the same arena.