The API to be used here is different in built-in render pipeline and URP. Setting `Camera.stereoTargetEye` in URP is not allowed, and there should be a warning when you try to set it with URP active. The warning isn't displayed correctly in your case when the property is set from a Start() event function - this will be fixed for 23.3.
The corresponding API to use with URP is `Camera.GetUniversalAdditionalCameraData().allowXRRendering`, which is what "VR Spectator Camera" asset store package is doing (note that this is not Unity/URP code) - this appears to be correct.
The confusion is likely caused by the fact that both of these fields are displayed in the UI with the label "Target Eye", but they correspond to different serialized data. This is due to backwards compatibility with built-in render pipeline.