Note that while the downsides you mention were naturally overcome by smartphones today, they're a) kind of expected to be improved on by progress of technology, and b) are completely orthogonal to the features of the PDAs which were not replicated in smartphones of today. Like, physical keyboards, stylus access, productivity-oriented OS.
Pieces of a perfect productivity tool are, or were, all there, scattered across time and devices. PDAs, feature phones, modern smartphones. For some reason, they were not put in a single package so far.
You are right that a lot of features weren't brought to current smartphones, just as current OS are not 'productivity' focused the same way windows is for instance.
I personally think it's because they just weren't worth it for the general public (including pros). Keyboards on android devices didn't convince, interestingly stylus and heavy multitasking are kicking and alive on the Samsung Note series, but only on the iPad for iOS.
'PDA's for specific, dedicated use cases are still a ton in the wild, just as custom android devices also flourish in the manufacture, retail or goods management world (basically "true" productivity-oriented worlds). They found their niche in that respect, the same way Google Glass also went to the business world in face of customer resistance. I see it as a natural evolution in that respect.
Pieces of a perfect productivity tool are, or were, all there, scattered across time and devices. PDAs, feature phones, modern smartphones. For some reason, they were not put in a single package so far.