So, I edited the PKGBUILD to compile everything in debug mode, which included simply replacing all occurences of "Release" with "Debug". Started Eddie with the shell command given in /usr/bin/eddie-ui with an addition from the docs and got two additional lines of output, two:
$ mono --debug /usr/lib/eddie-ui/eddie-ui.exe --path.resources=/usr/share/eddie-ui --path.exec="/usr/lib/eddie-ui" "$@"
. 2022.03.04 22:40:28 - Eddie version: 2.21.4beta / linux_x64, System: Linux, Name: Arch Linux, Version: Linux 5.16.11-zen1-2-zen #1 ZEN SMP PREEMPT Mon, 28 Feb 2022 10:17:33 +0000 x86_64 GNU/Linux,
Mono/.Net: 6.12.0 (makepkg/c621c35ffa0 Thu Jun 17 02:48:02 PM -03 2021); Framework: v4.0.30319
. 2022.03.04 22:40:28 - Command line arguments (3): path.resources="/usr/share/eddie-ui" path.exec="/usr/lib/eddie-ui" clr="--color=auto"
. 2022.03.04 22:40:28 - Raise system privileges
Elevated Debug: Waiting for client
Elevated Debug: Command:session-key
F 2022.03.04 22:40:32 - Unable to obtain elevated privileges (required): Unable to start (Elevated communication closed)
. 2022.03.04 22:40:33 - Shutdown in progress
. 2022.03.04 22:40:33 - Shutdown complete
I hope it helps!™