Due to security concerns I have turned off user visibility. My understanding wa

Have you had a chance to look at group privacy policy? You can configure security type, privacy policy to yield expected results required.

forum.appian.com/.../Group_Privacy_Policy.html

In addition to that, there is a smart service Modify User Security can be used to set viewers on a user profile.

I recall testing visibility settings for groups when user visibility is limited. From my results, I recall not seeing the members of the group no matter what settings I set if user visibility was limited.

If you test this out on your side, you should be able to note the behaviors with each setting.

If you open the group up in /designer and go to the properties tab you should see a section for security settings. This will let let you modify the Type (who can see the group?), membership policy, and privacy policy.

Let me break up my questions and see what you all think. :)
I don't want a Basic User to go to Tempo - Records - Users and see everyone in the platform. I set DEFAULT_VISIBILITY to none on the platform where Users are only able to see their own User Record. But, this has broken some applications that are using UserPicker. How can I get it to they will see their User Record plus anyone else in the same group?

I believe that is where the problem will pop up. Impact in various areas of Appian with regard to user visibility from documentation you may have already reviewed:

forum.appian.com/.../User_Profile_Visibility.html

Within that documentation, is Impact on Search:

Impact on Search
If a user is not visible, the user does not show up in any search results, including the People Picker. As discussed earlier, the user can still be found through group membership lists, but only the username appears.

Also, impact upon users/groups:

User Lists
If a user is not visible, the user will not show up in generic user lists, such as the Users record list in Tempo or the initial list displayed when accessing the People tab. He or she will, however, still display in the membership list of any groups the user is a part of. In this case, only the user's username displays.

Besides the user visibility setting, the user's role map, which determines the specific visibility of a user to a set of other users.

I'd review the documentation there pretty carefully, and see if you can find anything useful to incorporate on your side to try and get the desired results.

Thanks all. I will just play around with these options in my dev region.