Shifted profile from one quota to another and it disappeared
Using Jamroom
Ok, found the reason. The profile_url is set off of the "Profile Name" but you've set the Profile Name to only be visible to users in the quota "(quota) Central Administration".
Since the profile 'indexing' is not in that group, it means it doesn't have a profile name. If it doesn't have a profile name, it doesn't have a URL.
I think I'm understanding what your thinking:
My guess is that you think the by setting the "Display Groups" to "(quota) Central Administration" that users who have a profile also in Central Administration will see that form field for THIS profile (indexing) which is in another quota.
So then the admin users could come along and update it because they also own a profile in the Central Administration quota. That's what my guess is.
If my guess is correct: then, no, thats not how it works. the 'Indexing' profile is in the "Indexing Hub" quota. So if you want any form fields to show FOR EVERYONE when they visit the profile editing page, you need to allow those fields in the Form Designer.
The exception is to use the (group) settings. They are user settings, not (quota) settings.
In either case though, if the profile name is gone the url is going to be cleared out. Which is not good.