The problem was caused by the fact that when you open emoji panel in Windows 11, a search edit field gets focused. This causes NVDA to treat arrow keys as text navigation commands. Telling NVDA to not treat this field as an edit field fixes the issue, which is the solution implemented in Windows App Essentials and in upcoming NVDA 2022.1.
As for Windows 10 emoji panel and the problem of this panel closing when you press keys, that's new to me - I may need a debug log to figure out what's happening.
As for NVDA assuming the role of Windows App Essentials as far as emoji panel support is concerned, not entirely. There is an issue with IME (input method editor) handling in Windows 11 that I'm still testing with Windows App Essentials add-on, and I plan to submit a pull request to NV Access to get this resolved - part of the delay has to do with Korean IME changes in Windows 11 which I'm still testing, potential changes with upcoming Windows 11 Version 22H2 (about to enter final testing by Windows Insiders), and trying to keep up with school (I'm at a point in graduate school where I need to let go of most of my add-ons to keep up with reading materials).