The irony of this is off the scale. I'm clearly bored as I know the answer to this but I'm intrigued what's going on here.
Brand new install of Windows Server 2019 evaluation. Only configuration change made is to turn off IE Enhanced Security Configuration. Open https://www.microsoft.com/en-us/edge and click Start Microsoft Edge. You get this prompt:
On clicking Allow you get this message:
The workaround is to download the standalone installer but what's going on here? Microsoft making it difficult to install new Edge on Server 2019 seems a bit like shooting yourself in the foot.
BTW - Chrome installs fine from their web page.
But why won't the installer launch from the website?
Yah, this is a tad frustrating. I ended up going to https://www.microsoft.com/en-us/edge/business.
You'll need to enable Javascript and such before IE will let you download the install.
How to install Edge on Windows Server 2019:
Open IE:
Then do what @michael said and go to: https://www.microsoft.com/en-us/edge/business
And, yes, this is asinine.
I had to add https://msedge.sf.dl.delivery.mp.microsoft.com to the trusted sites after following @NotMe's steps to make it work.
Microsoft edge browser need to install to fix this problem https://www.microsoft.com/en-us/edge/business/download