You're right, convergence may never happen. It requires cooperation from the manufacturers, or one big one to just make a seemlessly connecting bunch of stuff with well supported stable interfaces that they don't plan to change.
But it's all perfectly possible now. Any decent modern smartphone has enough power to store Gb of data, run an HD screen, and power reasonable looking games or office software. That's simply unarguable. And that tech is getting cheaper all the time.
It may be that the tech becomes so cheap that it's easier to just have a tablet in every room, and something with a bigger screen and keyboard so you can type properly.
But I doubt it. Because it's just as much effort to integrate all the software, so that you can get your stuff migrated on to all this stuff and set it up. Given most people are incapable of configuring their current devices properly. And the demands of software are still growing, plus this stuff takes building, and natural resources, and transport and sales costs. So there must be a minimum price somewhere, unless we get matter transformers. Or giant robot factories in the asteroid belt.
I already cast stuff from my phone/tablet to my telly. And to my speakers. I have separate PCs, but I don't game on them anymore, so my only requirements are for office, media and web browsing. All perfectly doable on phones and tablets . And a computer is still more expensive than a monitor. Although it could soon be that a £5 full PC on a chip is possible, so all screens are smart.
Losing your phone is admittedly a problem. Being portable it's at high risk of breakage and loss. But that's a piece of tech that's always going to need to be smart, as smartphones are just so useful. So it would probably still end up being cheaper to just have a spare, and whatever non-portable peripherals you feel you require. I suspect most people will be happy to do most of their personal computing on a tablet, whcih the phone could slot into, or could be smart. Then only work will require a keyboard and screen. And they'll either act as remote controls to things like media and games systems, or even be the system.
Specialist stuff will probably always be different, gamers will probably always want 10% more performance for double the price - but most people's computing needs are pretty modest.