So, the thrust of the question seems to be why we don't regard 'sitting' in 'Jane was sitting' as an adjective, and why is 'is sitting' considered a present continuous construction there, as opposed to a copular predicate adjective construction.
This is one of those times when it becomes very important to distinguish between three very different linguisticky things:
- grammatical relations
- word categories
Let's consider an example similar to the Original Poster's:
- Jane is entertaining.
This sentence is ambiguous between a reading where 'entertaining' is a verb and one where it's an adjective. The verbal reading is, further, ambiguous between one where Jane has guests right now and another where she is providing amusement or enjoyment to some being(s) or other. In order to keep the meaning of the verb and the adjective as similar as possible, let's just consider the latter case where the verbal reading means that Jane is amusing someone, and where who exactly is recoverable from the context.
Next notice that although the verb entertain has a dynamic meaning, the adjective does not. The verbal reading of 'Jane is entertaining' also means that Jane is entertaining now. Jane cannot be asleep in the verbal reading of 'Jane is entertaining' (i.e. 'Jane is entertaining [some people]'), but she obviously can in the adjectival one (i.e. 'Jane has this quality where people would find her amusing to be with'). In fact, in the sentence where 'entertaining' is an adjective, Jane could have been holed up in solitary confinement for the last decade, and currently be holed up still, unseen by anyone. While verbs can have stative or dynamic meanings (where they describe either states or actions), adjectives have stative meanings - even when they describe a propensity to act in a certain way, like the adjectives explosive or violent or loquacious. Those adjectives describe propensities or tendencies not actions themselves.
We must come to the conclusion, therefore, that there is a semantic difference between the construction where entertaining is an adjective and the one where it is a verb.
The Original Poster discusses cases where they believe that 'sitting' is an adjective, for example when it occurs as an attributive modifier of a noun. In fact, occurring as an attributive modifier of a noun is a grammatical relation not monopolised by adjectives, but shared by verbs as well as nouns themselves and even determiners.
Consider the following example:
This could mean either a husband who generally has the propensity to amuse (adjective), or a husband who is in the process of entertaining people or hosting people (verb):
- the entertaining husband should provide canapés
Similarly, the verb 'sitting' describes people who are currently in a state of sittingness, or who currently carry out the action of sitting: 'the sitting president' or 'a sitting duck'.
However, meaning is a tricky and often unreliable way to distinguish parts of speech, or categories of phrase (notice that OP's original reasoning seems to have a semantic element, in that they argue that the verb sitting describes the subject Jane in the same way that nice would in a copula construction). A much better way is to carry out syntactic tests. Some of these have been demonstrated by @BillJ in his answer here.
We can complement BillJ's tests with some more. Verbs sometimes take direct objects, in other words they can take noun phrases (NPs) as complements. Only a couple of adjectives, one of which is the adjective worth, can take an NP complement. So in the following sentence entertaining must be a verb, not an adjective:
- Jane was entertaining the audience.
Also many adjectives can be pre-modified by the comparative adverb more, whereas verbs cannot:
- Jane was more entertaining than Bob.
- *Jane more entertains than Bob. (ungrammatical)
I'd like to move the discussion up a gear here. So let's distinguish three types of construction with the auxiliary verb be (we'll be using auxiliary verb to refer to a class of verbs, not an incidence of a case of be being followed by another verb):
- be dismantled (passives)
- be arriving (continuous constructions)
- be nice (copular constructions)
Let's call the different bes here passive be and continuous be and copular be respectively.
So, one way that we can show the copular and continuous constructions to be different is that a copular be verb phrase can function as the complement of a continuous be:
- Jane was being entertaining.
This can't work the other way round:
- *Jane was entertaining be being. (ungrammatical)
So, it seems that this is clear evidence that the two constructions are not equivalent. One can be embedded as part of the other but not vice versa.
Lastly, why consider the [be + present participle] construction an aspect?
An aspect is usually an inflection on a verb which indicates how the state or action it denotes is being represented/viewed in time. Clearly the construction under consideration is not an inflection of a verb. However, we can also consider so-called periphrastic aspect, which is when a multi-word verbal construction is used to convey the same kind of meaning. This is similar to the concept of periphrastic tense. Tense is usually considered an inflection on a verb which characteristically is used to refer to present, past or future time. However, some linguists conceive of the perfect construction in English as a periphrastic past tense. Or the use of will + verb as a periphrastic future. [I don't subscribe to these ideas about periphrastic tense in English, myself. That doesn't count for much.]
Well, it does indeed seem reasonable to consider the [be + present participle] construction a periphrastic aspect, where a combination of the auxiliary be and a present participle -ing suffix on the lexical verb serve to indicate that the state or action denoted by the verb can be considered as being concurrent with another time (which can cause all kinds of implicatures - for example that this state or action is/was/will be only temporary).
It is further reasonable to conceive of the construction in this way because it can combine with all kinds of other verbal constructions: ones with modal verbs, present tense auxiliaries, past tense ones, perfect constructions and so forth. And in each of these cases, the aspectual meaning conveyed by the [be + present participle] is the same. And this is the same kind of job that is done by aspectual inflections on verbs in other languages.
And of course, this meaning and the meaning conveyed by a simple copular construction are utterly different. In fact in a copular construction, the presence of the verb be adds nothing to the meaning really apart from carrying the features of tense and person.
It is a worthy speculation, then, that the -ing form in a present continuous construction might be a predicate adjective. However, the result of any rigorous investigation will show that this is not the case. Is the present continuous a periphrastic aspect? Well, no. But it's reasonable to consider the continuous construction [be + present participle] a periphrastic aspect. But if it is, like aspects that occur as verbal inflections, it combines with different tenses and modal constructions.