SQL server permissions during installation
I will say up front that I do not know why your company would have policies that lock down the OS so much that a Microsoft product cannot be installed and provided the appropriate permissions during an installation. I would think they might be doing that backwards. I supported the military as a contractor for many years and that included secured networks, we did not lock down the OS until after all software was installed on the server. We at most wanted to ensure the software installed itself appropriately without any issue.

Outside of that, it is going to depend on what permissions you are not allowing to be set. If it is being restricted on what OS level permissions are set within the local group policy that are going to prevent the services from starting the installation will likely fail.

Shorting SQL Server services the documented permissions required to operate might affect your supportability with Microsoft. Just something to keep in mind.

Edit/Update

There is no in-depth doc that is going to go through what all the installation process does on a server. You can weed through the installation log under the default directory here: C:\program files\Microsoft SQL Server\110\Setup Bootstrap\Log. The 110 would be based on which version of SQL Server you are working with: 100 = SQL Server 2008/R2; 110 = SQL Server 2012; 120 = SQL Serve 2014.

I don't know anyone has gone through and determined the outcome of each little affect of an OS level permission not existing. All that is needed to understand is that permissions assigned according to documentation show what is required for a supported installation of SQL Server. You are basically going to experience sporadic issues based on what you need to do. Every permissions has a purpose, you just may not have reached or hit that purpose as of yet.