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At an elementary level, often, Japanese learners are taught that です is equivalent to the verb "to be" in English. Typical example:

私は学生です, I am a student.

That's fine, after all, it works. And to be honest, most people probably wouldn't really need a different explanation for the rest of their life.

With this question I'd like to go a little deeper. Can we really call です(or だ) an auxiliary verb?

To start with, if you look だ up on goo, it is grouped together with です, である, and のだ.

Right off the bat, you get this explanation:

断定を表わす。 expresses a conclusion/decision

So here is not really mentioning the word "verb" or "auxiliary verb" at all. Shortly after you can see:

「です」は、「でございます」が変化したもの,

So, we can start by saying that actually です is an alteration/contraction of でございます.

Continuing to read the definition we get:

「である」は、「で」に補助動詞「ある」が接続したもので、同様のものに「であります」「でございます」がある。

Hence, 「である」 is formed by attaching to「で」the subsidiary/auxiliary verb「ある」. Because actually:

「で」は「だ」の連用形。 「で」is the conjunctive/continuative form of 「だ」

Therefore, if I'm not wrong, we can do the following passages:

「で」+「あります」=「であります」= 「でございます」 = 「です」

Does the above make sense? In particular I'm a bit doubtful about the second equality.

To sum it up, 「です」 is a variation of 「だ」(its polite form). It is also a contraction of 「でございます」, which in turn is the same as 「であります」(forgetting here about the differences in usage), that is formed by attaching 「で」 to the verb「あります」.

So, after this long trip, we arrive to the conclusion that the "hidden verb" behind 「です」, is in fact ある, which is actually (or at least more close to) the sought after "to be".

Is this a correct explanation? How can it be improved? Or, if it's wrong, where and why?

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Your collection of questions conflate a few things: 1) what is だ・です in modern Japanese, and 2) how did だ・です derive historically.

Because of #2, #1 is a bit ... messy. :) So let's start with the history.

「です」は、「でございます」が変化したもの,

This isn't an explanation of what です is now, so much as an explanation of the historical derivation. Shogakukan's 国語大辞典 provides this description:

「でござります」→「でござんす」→「であんす」→「でえす」→「です」の経路で生じたものという

Meanwhile, the modern plain form だ derives as:

「にてあり」から出た「である」が、〔dea〕→〔da〕と変化したものという。→であ・じゃ・である

Modern で (which actually appears from around the 1300s-1500s) is understood to have evolved from earlier にて (which is still used in formal writing). Classical copular ("is") verb あり equates to modern verb ある.

Looking now at the particular parts of your post:

「で」は「だ」の連用形。 「で」is the conjunctive/continuative form of 「だ」

Well, yes, in the modern language. But historically, だ derives from で + ある -- so this is not a regular kind of conjugation paradigm.

To sum it up, 「です」 is a variation of 「だ」(its polite form).

Yes. Though I might not use the term "variation" so much as "polite form".

It is also a contraction of 「でございます」,

It derives from a contraction of でございます, but I don't believe it's considered to be such a contraction in modern regular usage.

which in turn is the same as 「であります」(forgetting here about the differences in usage)

It's not quite the same as であります. Note that ございます is the humble version of あります. That difference in usage is important. :)

that is formed by attaching 「で」 to the verb「あります」.

Yes -- but note also that であります is not a verb unto itself, but simply the particle で plus the verb あります.

So, after this long trip, we arrive to the conclusion that the "hidden verb" behind 「です」, is in fact ある, which is actually (or at least more close to) the sought after "to be".

Historically, yes. And if you dig around in classical Japanese and older stages of the language, you'll find just that -- あり serves as the primary copular verb.

In modern Japanese, however, だ・です is the primary copular verb: it closes a predicate, and (optionally) supplies social register information ("politeness"). One no longer says things like 綺麗にあり, one says instead 綺麗だ. Similarly, instead of 本にてあり, one says 本だ.

Related thread touching upon social register:

I hope the above covers the bases for you. If not, please comment and I can edit the post accordingly.

Addendum

I missed a question of yours that I'd meant to answer.

Can we really call です(or だ) an auxiliary verb?

Depends. :) "Auxiliary verb" (助動詞) is a kind of loose category in Japanese school grammars. The article on the Japanese Wiktionary provides a table showing the various things that get this label. Some of them I really don't agree with: そうだ and ようだ are listed here, and those are clearly そう or よう + copular だ・です -- and だ・です is right there in the table as it's own pair of rows at the bottom. Others like らしい, ない, and たい are adjectival in form and function. Many, like させる or たがる, have full conjugation paradigms (i.e. they have forms for all conjugation slots), whereas others like negative supposition まい or positive supposition う are defective (i.e. have incomplete conjugation paradigms; see Defective verb at Wikipedia).

Basically, it's a grab-bag of odds and ends that get stuck on the ends of other things (often, but not always, verbs). This category was also previously called simply 助け言葉 ("helping words"), and this vague moniker is a clue to the vagueness of the category itself.

So, "can we really call です(or だ) an auxiliary verb?" Sure. So long as we're clear on what we mean by "auxiliary verb". :)

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    Actually I think this is a great answer. Very helpful to clarify what my understanding was. I think I wasn't really far away from the right answer, and was looking just for some remarks like yours to put everything in the right place. It's quite a shame stuff like this is not covered in most of Japanese language books. Sometimes I have a feelings things are "simplified" just to get the reader to get the point.. but they should talk about this stuff somewhere. – Tommy Nov 5 '18 at 7:23
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    As a side note, I don’t believe it’s known for sure whether です is actually from でございます, or whether it’s from であります, でおわす, でする, etc. All of them exist(ed) side by side, and all of them tend(ed) to be paired down to shorter versions; which of them, if any one in particular, ended up being standardised as です is a highly complex question that may not have an answer at all. – Janus Bahs Jacquet Nov 5 '18 at 11:26
  • @JanusBahsJacquet, agreed -- I thought I'd read somewhere about であります as the basis for です, but when I went looking, I found only the でございます shift above, and a shift from でそうろう → でそう → です. Shogakukan's entry states that the latter was limited to 狂言, which seemed irrelevant here so I didn't add it to my post. From what little I've read about Japanese in the Edo period and earlier, でございます seemed to be used more in speaking than であります, making it more likely that speakers would abbreviate that construction. OTOH, language is nothing if not complex; です could be from "all of the above". ¯_(ツ)_/¯ – Eiríkr Útlendi Nov 5 '18 at 16:48
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    @Tommy, ya, it would be nice if more of this stuff were available (easily findably, anyway) in English. That said, any textbook explanation has to draw the line somewhere, or run the risk of drowning its readers in excessive detail and turning them off altogether. I'm personally fascinated by etymologies and historical shifts over time, and I'm spending (probably far too much) time over at Wiktionary, mostly adding etymologies to entries by translating the info I can find in Japanese-only sources -- precisely because I was stymied by the lack of same in materials for learners. Cheers! – Eiríkr Útlendi Nov 5 '18 at 16:54
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    Erik and @Tommy – if you’re interested in easily accessible Japanese etymology and don’t already know about it/own it, I can recommend Bjarne Frellesvig’s A History of the Japanese Language (Cambridge UP). Very useful and well-written book with a plethora of interesting details! – Janus Bahs Jacquet Nov 5 '18 at 17:03

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