AI insights for the tasks and a stand-alone application is an amazing idea !
I actually had a similar idea of creating a desktop application which will always stays as a sidebar in our OS on one side taking around 10% of the screen width and the rest of the applications will open in the remaining portion of the screen. This sidebar will be kind of like your assistant, to show the upcoming tasks, ongoing tasks with timer/progress bar, etc. Along with other OS properties like time, CPU usage, etc. And adding tasks and notes directly. Also drag and drop feature, since Obsidian is your second brain, you can directly drop the content you are going through from any application to Obsidian.
And using AI insights will even take this idea much further. Looking forward to further development on this project and possible contributing to it!
I have also developed a Task Management plugin, called Task Board. Integrating it with this application in a separate tab along with various other tabs, with different functionalities feels to be a very helpful tool.
My Task Board post : Task Board : Another GTD methodology | Similar to GitHub Projects Planning