Your explanation about when BE is not a linking verb is clear, that is, where its complement is an adverb. However, you still have not explained the following:
The leaves are everywhere [everywhere = adverb; therefore, are = not linking]

He was late [late = adjective, not adverb (per your view); therefore was = linking verb]
The two forms of the verb be above are quite different, that's why you can't call them the same way - the term non-linking verb is to provide you with the message that it differs from the normal linking-verb
...
To sum up my viewpoint:
The following is persuasive enough: (a) Where BE's complement is an adverb, then BE is not a linking verb;

(b) Deriving from (a), a linking verb can only have a nominative or an adjective as its complement.
Do you then agree that: (1) BE can have a predicate adverb as its complement, but in this case we cannot call it a linking verb;
Let's call it just a verb that links the subject with its complement, which (the complement) doesn't modify the subject as it would with the linking verb used; like I wrote above - the verbs differ (2) In "He was late", "late" is an adverb
adjective and "was" is not a linking verb.
it actually is 