According to Cobuild (Collins) (Chapter 1:7), the verb + -ing form catenation has three structures (not confusing the non-catenative strings such as Under the trees Bill strolled, looking at the flower beds):
[examples partly adapted]
[1] Verbs in phase (ie a two-verb structure where essentially one two-part concept is expressed):
The sea came rushing in.
He started / kept / stopped crying.
She avoided looking at him.
I won't bother going.
Have you tried asking?
I'm going shopping.
(These mostly invite echo questions such as 'What did he start doing?' 'What won't you bother doing?' With the 'going fishing' type, the echo question would be 'You're going ...?)
[2] Verb with object (ie with what is often termed a gerund)
I like being alone.
Have you considered applying?
She recommended staying.
He didn't remember leaving.
This involves stripping down the engine.
(These mostly invite echo questions such as 'What have you considered?' 'What does this involve?')
[3] Verb with adjunct ( depictive or resultative)
The soldiers died fighting.
Their boat finished up pointing the wrong way.
(These mostly invite echo questions such as 'How did the soldiers die?' (ie What was the manner of their death) 'How did the boat end up?')
I'd argue that these usages are [1] verbal, [2] verbal-nounal, and [3] adverbial or adjectival.
To determine whether auxiliaries are verbs, we should examine two kinds of properties. One kind of important property relates to word forms, and the other kind of property relates to word use.
Most verbs have properties such as tense, aspect and mode.
The verb to be is a normal, complete verb. Was and were are past tense forms of to be, and is, are and am are present tense forms. Being is a continuous aspect form, and been is a perfect aspect form. The were of "if I were a rich man" and the be of "be he live or be he dead, I'll grind his bones into my bread" are subjunctive forms.
If can is a verb, then it is a defective verb. It doesn't have forms that show aspect. The forms canning and canned don't exist for the auxiliary. Neither does the infinitive to can.* However, it does have the properties of tense and mode. Could is both a past tense form and a subjunctive mode form, just as were is both a past tense form and a subjunctive form of to be.
Word form isn't the only way to show a verb's mode. The interrogative mode is usually shown by the verb's position.
The statement "he is a student" employs the indicative mode. The interrogative mode places the first word of the verb** before the subject: "is he a student?" The statement "he can be a student" is subject to the same transformation: "can he be a student?"
A clause pairs a subject with a predicate. A predicate requires a verb. We consider a statement like "he studies" to be a complete clause. In answer to the question "is he a student?", the answer "he is" also counts as a complete clause. If, in answer to "can he be a student?", the statement "he can" is a complete clause, then the can must be a verb.
Although we can see that the auxiliary can doesn't exhibit every property that most verbs have, it does exhibit properties that only verbs have. It has a form that marks tense or mode. Its position can indicate a mode that word forms cannot. It can act on its own as the predicate of a clause.
All these reasons support the idea that defective verbs are verbs. We don't have a good reason to place such words in a different grammatical category.
_______________
* Yes, there are homonyms that do have continuous, perfect and infinitive forms. We won't consider those to be the same verb.
** Most one-word verbs require that a word be added to the verb phrase so that the added word can be moved. The statement "he studies" becomes "he does study" on its way to becoming "does he study?" The notable exception to this is when the one-word verb is a form of to be.
Best Answer
I think you are misunderstanding the meaning of finite. Finite is not a category of verb but a category of verb forms and uses.
Finite forms are those which must take either past or non-past tense (must is anomalous in having the same form for both tenses) and may change to agree with the person and number of their subjects. Non-finite forms, infinitives and participles, do not change with tense of the utterance or person or number of the subject.
In your example, the modal verb could is finite: it is the "past" form of the verb can. (It may not express past tense, but that is another matter.) As you say, it doesn't take a particular inflection which expresses person and number; but no English verb has a complete repertory of such inflections. You will find a little more information at this question.
In fact, the full modals can, may, must, shall, will differ from other verbs in being defective: they have only finite forms, no infinitive or participles.
Every complete clause has exactly one finite verb✲; if there are more verbs strung together, then the first is finite and the rest are non-finite. Consequently, if there is a full modal verb it must be the first in the string.
✲ Except in cases where two or more finite verbs are conjoined: I can and will do it. But these cases really express two or more clauses.