Put a default value list value based on the current users

Is there a different way to accomplish this goal?

If I understand you correctly, you don't need the Person object to achieve your goal. You don't even need a concatenated placeholder, either.
With the configuration shown in your screenshots and what you want to achieve I would proceed in the following way:

1.) in M-Files Admin, add the property called "M-Files user" to your "Member User" Class
2.) in M-Files desktop, populate this newly created property for all "Member User" objects manually (don't know if there is any automation procedure for that, I did it manually since we are a very small company). Alternatively - for testing purposes - populate only a few.
You now have created a connection between two technically entirely different levels - an Object in M-Files and the corresponding M-Files user, simply by adding the M-Files user as a property to your "Member User" Object.
3.) in M-Files Admin/ Metadata Card Configuration, you now have to create a Custom Placeholder called, for example, CurrentUserMemberUserObject. (go about just as you did before, but your reference should not be the Person object, but the MemberUser object). Configuration should then be:
Name = CurrentUserMemberUserObject
Object = Member User
Property = M-Files User
Mode = Static Value
Static Value = (Current User)
4.) Still in Metadata Card Config, populate the property "Member User" within "Test Class" with a dynamic value. The placeholder to choose is the one created in step 3. No concatenation nor anything else needed. Just the placeholder.

This should do the job. Now if you create a new object belonging to Test Class, the "Member User" property should default to the current user. If I got you correctly, this in turn will automatically fill your "Member" property, which, in turn, will set the right permissions.
Feel free to ask if anything remains unclear.

Is there a different way to accomplish this goal?

If I understand you correctly, you don't need the Person object to achieve your goal. You don't even need a concatenated placeholder, either.
With the configuration shown in your screenshots and what you want to achieve I would proceed in the following way:

1.) in M-Files Admin, add the property called "M-Files user" to your "Member User" Class
2.) in M-Files desktop, populate this newly created property for all "Member User" objects manually (don't know if there is any automation procedure for that, I did it manually since we are a very small company). Alternatively - for testing purposes - populate only a few.
You now have created a connection between two technically entirely different levels - an Object in M-Files and the corresponding M-Files user, simply by adding the M-Files user as a property to your "Member User" Object.
3.) in M-Files Admin/ Metadata Card Configuration, you now have to create a Custom Placeholder called, for example, CurrentUserMemberUserObject. (go about just as you did before, but your reference should not be the Person object, but the MemberUser object). Configuration should then be:
Name = CurrentUserMemberUserObject
Object = Member User
Property = M-Files User
Mode = Static Value
Static Value = (Current User)
4.) Still in Metadata Card Config, populate the property "Member User" within "Test Class" with a dynamic value. The placeholder to choose is the one created in step 3. No concatenation nor anything else needed. Just the placeholder.

This should do the job. Now if you create a new object belonging to Test Class, the "Member User" property should default to the current user. If I got you correctly, this in turn will automatically fill your "Member" property, which, in turn, will set the right permissions.
Feel free to ask if anything remains unclear.