I think the "cloud" is helping to make it seamless to go from iPad to desktop to phone with one work item (like a report for work), but it's not quite there yet, I agree.
(I think in the Star Trek universe you would just reclaim the PADD in the replicator once you're fully done with it rather than move it around to some central "PADD library", right?)
Within an ecosystem (MS-only, Apple-only) it sorta works; much worse, if you need something from outside, then it doesn't.
RE replicators, you're right! I don't know why I haven't thought about it. PADDs are very simple as electronics go (even as today's standards go). DS9 showed people recycling dishes and food through a replicator, so the same would work for PADDs.
(I think in the Star Trek universe you would just reclaim the PADD in the replicator once you're fully done with it rather than move it around to some central "PADD library", right?)