I was able to get an autologon KIOSK to autolauch Edge and pin it to the Start menu. Now I just have to figure out the behavior of all the Edge arguments. I applied this XML through a custom configuration profile in Intune.
<?xml version="1.0" encoding="utf-8" ?>
<AssignedAccessConfiguration xmlns="http://schemas.microsoft.com/AssignedAccess/2017/config" xmlns:rs5="http://schemas.microsoft.com/AssignedAccess/201810/config" xmlns:win11="http://schemas.microsoft.com/AssignedAccess/2022/config">
<Profiles>
<Profile Id="{9A2A490F-10F6-4764-974A-43B19E722C23}">
<AllAppsList>
<AllowedApps>
<App DesktopAppPath="%ProgramFiles(x86)%\Microsoft\Edge\Application\msedge.exe" rs5:AutoLaunch="true"/>
<App AppUserModelId="Microsoft.Windows.Photos_8wekyb3d8bbwe!App"/>
<App AppUserModelId="Microsoft.BingWeather_8wekyb3d8bbwe!App"/>
<App AppUserModelId="Microsoft.WindowsCalculator_8wekyb3d8bbwe!App"/>
</AllowedApps>
</AllAppsList>
<win11:StartPins>
<![CDATA[
{ "pinnedList":[
{"desktopAppLink":"%ALLUSERSPROFILE%\\Microsoft\\Windows\\Start Menu\\Programs\\Microsoft Edge.lnk"},
{"packagedAppId":"Microsoft.Windows.Photos_8wekyb3d8bbwe!App"},
{"packagedAppId":"Microsoft.BingWeather_8wekyb3d8bbwe!App"},
{"packagedAppId":"Microsoft.WindowsCalculator_8wekyb3d8bbwe!App"}
] }
]]></win11:StartPins>
<Taskbar ShowTaskbar="true"/>
</Profile>
</Profiles>
<Configs>
<Config>
<AutoLogonAccount/>
<DefaultProfile Id="{9A2A490F-10F6-4764-974A-43B19E722C23}"/>
</Config>
</Configs>
</AssignedAccessConfiguration>