Your monitor’s screen resolution will impact the available settings and external screen capabilities.
While undocumented, testing has demonstrated that 4:3 monitors with a display resolution of 1280x1024 and below are currently only usable with iPad screen mirroring; as such, associated settings to disable screen mirroring in favour of Stage Manager are not available in iPad Display settings.
When used with a 16:9 widescreen monitor, with a native display resolution of 1920x1080 or greater, the full set of Display settings - and access to Stage Manager - become available.
The iPad User Guide lists the upper resolution limit as 6K - but the Guide offers no lower limit, or suggestion of aspect-ratio, being a factor for compatibility:
Connect iPad to a display with a cable - Apple Support
This discussion thread may be of additional interest:
iPad Extended Desktop not working - Apple Community
In conclusion, your monitor must be capable of 16:9 widescreen 1920x1080 (FHD) to support iPadOS16 Stage Manager extended desktop. Unless the monitor meets the minimum required specifications, you will not see the full range of controls in iPad settings.
Whether or not this is an intentional limitation, or a bug introduced with release of iPadOS 16.4.1, is unknown.
