While in some specific cases, this may be true, it’s not the case that this can be automatically used to develop a populace badge for any group’s arms.
For an example, let’s look at the West Kingdom’s arms, and try to use the “remove the wreath and crown” method for making a populace badge.
First, we find the Kingdom’s arms. That’s easy enough, and we get <Or, within a laurel wreath vert, a crown of three points voided, vert.> Next, we remove the crown, <Or, a laurel wreath vert.> And finally, we remove the wreath… <Or.>
If this were the Western Realm’s populace badge, no-one could ever wear yellow without being seen as affiliated, and that’s clearly a problem.