Vital in a good way or a bad way? I.e., is the modification required for or detrimental to the re-installation?
You're making a great argument for a requirement that the modification be documented permanently within the equipment.
Obviously, replacement of a bad plug with a functionally-equivalent new one doesn't alter the functionality of the cord.
My hypothetic was attemtping to describe a scenario where the modification may leave the equipment suitable for the present application but may make it unsuitable for a different one.
Not only that the equipment modification is undocumented but it also opens WIDE and without prequalifications who can judge the equipment's suitability for the application. IMO the NRTL labels should identify the equipments modes of use and the labeling should also mean that the installation instructions are integrap part of the label. The AHJ's role should be to verify the the equipment is installed in accordance with labels provisions for use and that the installation instructions were followed.
The NRTL labeled instructions should be part of the installation documentation and should be permament part of the documentation that is required to be retained as part of the installations operating permit. I think that the NRTL label indicates that the equipment is built according to National Standards and restrictions on the acceptibility of those should be specifically overridden by local written codes ONLY if it offers a suitable and detailed technical alternative. The NEC should require minimum documentation to aid the AHJ's in their work.(It already does in certain aspects, but should go further.)