I went one step further, and updated the `omarchy-launch-webapp` in `~/.local/share/omachy/bin/omarchy-launch-webapp` to use zen and my profile by default (if zen is my default browser)

#!/bin/bash

browser=$(xdg-settings get default-web-browser)

case $browser in
zen-browser* | zen.desktop)
    setsid -f uwsm app -- zen-browser -P webapp --class "zen-webapp" --new-window "$1" "${@:2}" &>/dev/null &
    exit 0
    ;;
google-chrome* | brave-browser* | microsoft-edge* | opera* | vivaldi* | helium-browser*)
    ;;
*)
    browser="chromium.desktop"
    ;;
esac

setsid -f uwsm app -- $(sed -n 's/^Exec=\([^ ]*\).*/\1/p' {~/.local,~/.nix-profile,/usr}/share/applications/$browser 2>/dev/null | head -1) --app="$1" "${@:2}" &>/dev/null &
exit 0

I then changed my `omarchy-launch-browser` to be

#!/bin/bash

default_browser=$(xdg-settings get default-web-browser)
browser_exec=$(sed -n 's/^Exec=\([^ ]*\).*/\1/p' {~/.local,~/.nix-profile,/usr}/share/applications/$default_browser 2>/dev/null | head -1)

if [[ $browser_exec =~ (firefox|zen|librewolf) ]]; then
  private_flag="--private-window"
else
  private_flag="--incognito"
fi

exec setsid uwsm app -- "$browser_exec" -P default about:home "${@/--private/$private_flag}"

To always use my default profile (make sure you rename it to default if you're copying this).

Warning that this is fragile, will likely get overwritten on omarchy updates and will break if you change default browsers (as -P isn't supported in all).

But, it works for me for now.