1) Convergence works great for "casual" apps - messaging, stores, content feeds, etc.; anything where the amount of stuff you need to see on screen at once doesn't need to be especially large. But the web already does a great job of making this type of application both responsive and cross-platform.
2) Advanced, dense professional tools will never adapt automatically to mobile screens in any meaningful way. And frankly, these are all I find myself installing natively on my desktop anymore.
The vast majority of software these days falls squarely into one of these two camps, and neither seems to benefit much from the prospect of convergence.
I'd argue this is where Convergence actually works best. (Especially the way Windows does it, anyway).
For most people, "casual" apps are 95% of what they need a computer for. And the web does a good job, but not a great job, at making these apps responsive and cross-platform. Usually, these applications sacrifice lots of performance and/or usability and/or reliability, just for the benefit of being able to run in a browser. Which is fine if that's a trade-off you want to make, but this is usually not being done out of benefit for the user.
Convergence can help fix part of this, by reducing the cost of native app development even further while preserving the majority of the quality/performance/usability benefits native apps provide.
I would say the web does an adequate job, not a good job, at this.