Are modal verbs and auxiliary verbs actually verbs?

A friend recently told me that "can" is a rare verb without an infinitive. I have since looked it up and discovered it is an auxiliary verb. In my mind it modifies a "proper" verb in much the same way an adverb does.

"I can jump puddles"

"I could do it"

"to jump" and "to do" are OK, "to can" is wrong (meaning "to be able to").

"I am able to jump puddles"

"I was able to do it"

So, are words like "can/could", "must", "will/would", etc. actually verbs or are they only auxiliary or modal verbs and not "proper" verbs? I was also chatting to a Frenchman and he said in French auxiliary verbs are just called auxiliaries and are not considered to be verbs, so is it the same in English? It seems to me that if you can't "to" or "-ing" a word it is not proper verb.

"to jump", "jumping" - OK

"to must", "musting" - wrong

However, if auxiliary and modal verbs are actually verbs is there a word to describe a non-modal, non-auxiliary verb?

Please, no comments about canning factories or making wills. I already know that the words "can" and "will" can be infinitives but with a different meaning.

EDIT

Black bears, brown bears and polar bears are actually bears. Koalas and pandas are not bears yet some people call them koala bears and panda bears because they look a bit like bears. Likewise jellyfish and starfish are not fish but are aquatic animals as fish are. Alternative names for them are sea jellies and sea stars. Ironically, the seahorse is actually a fish. Things are not always what they're called. So are auxiliary verbs actually verbs or just called verbs?

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. 

The English modal auxiliary verbs

• can, could, may, might, shall, should, will, would, must, and sometimes need and dare

are, indeed, defective verbs in that they no longer inflect for anything and have become part of the machinery. If you want to call them Aux instead of Vb, you're within your rights.

However, then you have the problem of whether be is a verb. It turns out that be is always an auxiliary for some kind of construction: Passive, Progressive, Predicate Adjective, Predicate Noun, and others like

• He is to leave tomorrow.
• It's a long way to Tipperary.
• There's a man looking for you.
• What I need is love.

Unlike lexical verbs, the behavior of be is completely governed by grammatical rules, not by its nonexistent "meaning". Also unlike modal auxiliaries, which do have meanings, be has none, even though it inflects for everything, even first person singular present tense (am) and first and third person past singular (was and were; no other verb distinguishes plural from singular in the past tense).

So if you want to say an auxiliary is not a verb, then be isn't a verb, though it has many verb-like characteristics. One of them is that it can be in the verb chain, that line of auxiliaries before you get to the main verb at the end. In fact, be has two places in the verb chain -- once to mark Passive

• He was arrested.
• He has been arrested.
• He must have been arrested.

and once to mark Progressive

• He was skating.
• He has been skating.
• He must have been skating.

which can occur together, though not often, because the be's bump into one another
(marked with ?? below)

• We are being photographed.
• ??We have been being photographed.
• ??We must have been being photographed.

Note that the longest verb chains start with a modal auxiliary like must; that's because modals are the first auxiliary when they appear. That's the result of their being defective verbs without other forms. Be, have, and other auxiliaries require infinitives or participles to follow them; since modals don't have such non-finite forms, they can precede other auxiliaries, but they can't follow them.

If you were to ask my opinion, I'd say Auxiliaries are a variety of Verb. Why not?