Yes, the problem in the given scenario is that there is a certain number of specific employees who are saying they are working from home but are doing shopping, taking the kids to school and doctor, watching TV, running errands, surfing the net and so forth, never getting around to actually doing any work.
As you point out, the "approval method" is a elaborate attempt to hide the fact that unproductive employees ABC are now not allowed to work from home and productive employees DEF are still allowed to. Rather than communicate this directly to persons ABC, an elaborate system of misleading emails is constructed.
A more effective and less damaging way to handle this is to meet individually with the specific persons who are having problems and explain it has become clear they are not productive when working from home, and so they'll have to work from their office from now on. And if that doesn't work out, then they can be dismissed.
Instead of handling this situation in the obvious manner, which would be effective and would not publicly shame anyone, we have two emails proposed to be sent to the entire company establishing a global policy to deal with a minority of people who are having problems. The wording doesn't matter, the problem is that the email is sent to everyone to deal with a specific problem with specific employees.
This is done this way because management is comprised of ineffective cowards who are terrified of confrontation and communication.
These emails show that the real problem here is highly ineffective management. To improve things management needs to change their ways or get out.