What is true of words denoting one sort of concept is not always true of words denoting other sorts of concepts, even if they are generally considered to have the same 'part of speech'.
For instance, the progressive is X-ing construction sounds just fine when used with some verbs (John is running, Kate is writing), but odd, if not downright ungrammatical, with others (*Melinda is believing his story, *Steve is knowing the answer).
The word is is likewise compatible with the progressive only in some contexts: both José is silly and José is being silly sound okay (albeit not equivalent), but between That is my car and *That is being my car, only the former is acceptable.
In a broad sense, we can categorize words into three semantic groups: words referring to objects, words referring to properties, and words referring to actions. While these three groups generally align with the commonly used parts of speech terms noun, adjective, and verb, an approach based on semantics and actual usage rather than prescriptive rules makes a finer-grained classification possible.
Just as we have three basic semantic groups, we might propose three basic sorts of usage. We may want to make outright reference to a concept, or we may want to invoke one concept to modify another, or we may want to predicate (or assert) a concept.
The three semantic categories are not limited to specific sorts of usage, but there is clearly one sort of usage where each is best-suited, or prototypical in a linguistic sense.
Objects, like DOG, TABLE or CAR, are usually time-stable — they persist from moment to moment in a way that properties (e.g., SHORT) and actions (e.g., JUMP) do not. They exist out in the world and can usually be counted. They're prototypical referents. This prototypicality is reflected in the wider range of grammatical options available to them in the reference function. We can refer to property words and event words, but we have to mark them explicitly. We have to add a noun-making suffix -ness to refer to the property word dark (Darkness is unpleasant), and the suffix -ing or the infinitive marker to to refer to the action word run (I like running / I like to run). We don't need any such modification for more prototypical object words — in fact, there are additional linguistic strategies available when referring to object words: we can make them plural, for instance, or use articles. Compare I like dogs with *I like darknesses or *I like runnings — only the first sounds okay. Again, while we can usually make a plural of a non-prototypical referent, it takes more effort, and the meaning won't necessarily be the same: I like several sorts of darkness.
Properties, like YOUNG, FAT, BLUE, likewise show a preference for one sort of construction: modification. No special marking is necessary to indicate that a property word is serving a modifying function: the red car, the young girl. Object words and action words can be used for modification, but they need to be marked for that function by additional linguistic form. For example, possessive 's is one way to mark an object word for modification (the man's car), while -ing can signal that an action word is modifying a referent (the sleeping girl). In both cases, words denoting non-prototypical properties need extra marking. (In linguistics this sort of extra marking is called 'structural coding'). Similarly to how only more prototypical object words can inflect for plural number in English, more prototypical property words also have additional 'inflectional potential' available to them when serving a modifying function: they can inflect for degree through suffixation: young, younger, youngest, fat, fatter, fattest, blue, bluer, bluest (or through suppletion: good, better, best). Less prototypical property words can still mark the comparative, but they have to do it with a separate word: nuanced, more nuanced, most nuanced, not nuanced, *nuanceder, *nuancedest (stated another way, these property words demonstrate the greater inflectional potential of property words in a modification function, but at the same time they need greater structural coding to express that potential). Object words and action words used for modification can't express this inflectional potential at all: *the man's-est car, *the car most of the man, *the sleepingest girl.
Considered in this light, the word is is a verb in that it is part of a predicating construction, but both its function and its range of 'inflectional potential' and 'structural coding' differ greatly depending on the semantic class of the word it occurs with:
When it occurs with more prototypical action words (i.e. 'verby verbs'), it indicates the progressive (John is running) and may be used to mark the passive voice (That book was read by Maria). It may also be left out entirely; when that happens (e.g., John runs), the construction denotes a regular or habitual action.
When it occurs with less prototypical action words (i.e., 'not-quite-as-verby verbs'), particularly verbs denoting mental processes or mental states, it can't indicate the progressive but it can be used to mark the passive: *Jake is liking his new apartment sounds strange, but The answer was known by Tanya sounds okay.
When it occurs with property words (i.e., 'adjectives'), it may be used with or without progressive marking, but be is really the only word with which it can be used: Marek is silly means that we're ascribing him a general characteristic (i.e., it's not necessarily that his behavior in the present moment gives that impression), while Marek is being silly means that we're making a statement about his actions right now. Properties which are construed as 'permanent', 'inherent', or 'involuntary' are difficult to use with a progressive construction: *Marie is being French, *Alex is being hungry, and *Zendaya is being tall all sound awkward at best. The passive construction is also not available to property predicates: *Silly was been by Marek sounds very ungrammatical. When property words are predicated, they are marked with greater structural coding which action words don't require. The word is is that extra coding. Note, though, that when used in predication property words are able to express some of the inflectional potential of action words: the progressive is possible, but the passive isn't.
Finally, when used with object words (i.e., 'nouns'), the progressive generally isn't allowed: *This is being Tom, *John is being a doctor. In these constructions, the word is is really serving as the greater structural coding which is necessary to predicate an object word. Unlike predicated property words, however, predicated object words don't really have access to any of the inflectional potential of predicated action words. In other words, by these criteria we'd be justified in saying that 'properties are more action-like than objects', or 'adjectives are more verby than nouns'.
We can even use the relations between these semantic groups and functions to describe a conceptual space (this diagram is from William Croft's Typology and Universals, p. 187). In this diagram, the cells along the upper-left-to-lower-right diagonal express the most prototypical combinations of semantics and usage, and the cells furthest from that axis express the least prototypical combinations.
REFERENCE MODIFICATION PREDICATION
object ______object ______object
OBJECTS reference modifier predication
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PROPERTIES reference modifier predication
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action _____action _______action
ACTIONS reference modifier predication
What it all comes down to, I suppose, is what your definition of is is. For me, neither 'is is a verb' nor 'is isn't a verb' does a good job of describing what the word does.
(For a much more detailed treatment of this approach to language, see the works of William Croft, particularly the monographs Typology and Universals and Radical Construction Grammar.)