Service roads can be part of the public network — for example, many narrow public residential roads are mapped as service=alley in many countries (another OSM ambiguity). That said, I understand the hesitation, and why some might prefer highway=residential. Honestly, I wish we had a highway=quaternary tag (which is effectively how unclassified is used here locally) so that unclassified could be used for any small public road that doesn’t fit other definitions.
Your thought is reasonable in terms of the hierarchy of roads, and existing (mis)use. Unfortunately, it has been named, defined, and understood as in-residential, not residential-like. Ideally it would have been eg the standard =local road terminology, and =unclassified be =minor if not too British.
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