When you try Page Layout Preview As in Salesforce and get Insufficient Privileges, check the below things

1. Check whether your profile has CRUD permissions for the object.

2. For Delegated Administrators, add the Custom Object in Custom Object Administration section. Check this link for setting Custom Object Administration.

https://www.infallibletechie.com/2017/06/how-delegated-admin-can-set-field-level.html

Note:
Picklist fields will show all the values since the record type is not taken into consideration.