
From my understanding MicrosoftEdgeUpdate.exe updates the browser as well as itself. There are two scheduled tasks under the root of the Task Scheduler library, which are responsible for check regularly for updates
- MicrosoftEdgeUpdateTaskMachineUA
- MicrosoftEdgeUpdateTaskMachineCore
If you run these manually and then go and look in C:\ProgramData\Microsoft\EdgeUpdate\Log you will see the Edge update process looking for updates for the Browse and the Update engine.
This is what a browser check looks like
[MicrosoftEdgeUpdate:msedgeupdate][15456:2732][CheckForUpdate][app_id {56EB18F8-B008-4CBD-B6D2-8C97FE7E9062}][cdp_name msedge-stable-win-x64][version 90.0.818.46][nextversion ]
This is what an update engine check looks like
[MicrosoftEdgeUpdate:msedgeupdate][15456:2732][CheckForUpdate][app_id {F3C4FE00-EFD5-403B-9569-398A20F1BA4A}][cdp_name msedgeupdate-stable-win-x86][version 1.3.143.57][nextversion ]
The log entries immediately after each of those will tell you whether it found an update and if there was any issues downloading the update.