"We are able to exchange knowledge, beliefs, opinions, wishes, threats, commands, thanks, promises, declarations, feelings – only our imagination sets limits. We can laugh to express amusement, happiness, or disrespect, we can smile to express amusement, pleasure, approval, or bitter feelings, we can shriek to express anger, excitement, or fear, we can clench our fists to express determination, anger or a threat, we can raise our eyebrows to express surprise or disapproval, and so on, but our system of communication before anything else is language. ... it is a system of communication based upon words and the combination of words into sentences." "What is language?" PDF.
http://www.uio.no/studier/emner/hf/ikos/EXFAC03-AAS/h05/larestoff/linguistics/Chapter%201.%28H05%29.pdf
The number of sentences in a language is infinite. But every language has a mechanism that enables human beings to utter or understand an infinite number of sentences using a finite number of building tools.
To my way of thinking, knowing what those building tools are is of paramount importance for every learner/teacher of the language because they can help us in many ways.

How can we find them?

According to my own very simple theory:
Everybody knows that Past, Present, Future mean TIME.
TIME is an extralinguistic category. That is, it exists independently of both Life and Language. If it is really so, then the hierarchical structure of the main concepts looks like this:
TIME > LIFE > LANGUAGE.
TIME created LIFE and is superior to it.
To retain its superiority/= control over LIFE, TIME has three key features
placed inside LIFE. They are PAST, PRESENT, FUTURE.
Being the creator of LIFE, TIME is also in charge of it.
The three key features are not only to keep control, they are to keep
everything in good order (to avert chaos)/= to help LIFE as well.
The implication is that LIFE is free to do whatever it wants to, but everything that
happens in LIFE must obligatory be PAST, PRESENT, FUTURE.
The three key features are property of TIME, they are untouchable for anything
that belongs to LIFE.

LIFE created LANGUAGE and is superior to it.
Now, does it not seem logical to assume that (likewise): to retain its superiority/= control over LANGUAGE, LIFE must have some key features placed inside
LANGUAGE (to keep everything in good order to avert chaos)?

If the premise is true, we are sure to find those key features placed by LIFE inside LANGUAGE. It is also logical to expect that like Past, Present, Future are everywhere in Life, they are everywhere in LANGUAGE.

If what I am saying makes sense, we only have to answer this one question:
WHAT IS IT THAT WE CAN FIND IN ABSOLUTELY ALL ENGLISH SENTENCES?

1. It is far from being what you call “cleaning up the terminology.”
2. I absolutely agree with everything you are saying here.
Plus a couple of interesting PDFs:
Bridging the Grammar Gap: teaching English grammar to the iPhone generation 1
BAS AARTS, DAN CLAYTON AND SEAN WALLIS
University College London
http://www.ucl.ac.uk/english-usage/staff/sean/resources/bridging-grammar-gap.pdf

The Future of English?
A guide to forecasting the popularity of the English language in the 21st century
David Graddol
https://microsites.bournemouth.ac.uk/business-postgraduate/files/2014/09/CIGBE-The-Future-of-English.pdf

There are a couple of things that make me feel a little uncomfortable, too:
1. My students (Russian speaking adults and teens) say that for them listening without understanding is much more boring than learning grammar.
2. We are sure to see a new generation of English learners, but will we see a new generation of English teachers?

I think in your example with a foreign teenager spending two years in England, it is mostly a matter of time and money rather than any teaching methodology.

Unless Warsaw Will finds a way to deny me access to his blog, I’m afraid it’s going to be for very long that I have little time left for something else.

If I had read “The twelve tense system in English - an overview” http://random-idea-english.blogspot.com/2013/08/the-twelve-tense-system-in-english.html thirty
years ago, when I really felt nothing but “frustration with verbal structures in English”, I think, I wouldn’t have had to start looking for something else as an alternative to “tenses”. But I hadn’t, so I had to.

By the way, there is one more work on the tenses (US English) I like and feel attracted to:
http://www.chabotcollege.edu/languagearts/esl/verb%20tense%20book--pdf.pdf

Anyway, that English can do perfectly well without “Tenses” having been declared,
perhaps there is someone who’s expecting me to explain in more detail what I mean by that.

TERMINOLOGY

1. The same all well known words like Sentence, Verb, Mood, Aspect, etc., except
that “tense” is not mentioned anywhere in their definition/description/labeling.
For example, there are no things like:

“While aspect always includes tense, tense can occur without aspect (David falls
in love, David fell in love).”
http://www.ucl.ac.uk/internet-grammar/verbs/tense.htm
or
“For each grammatical tense, there are subcategories called aspects. Aspect
refers to the duration of an event within a particular tense.”
http://www.elearnenglishlanguage.com/blog/learn-english/grammar/tense-aspect/
or
We are waiting for the bus. = Present Continuous Tense

2. As a follow-up to the above, I am inclined to think that MOOD, VOICE, ASPECT,
TIME are not grammatical categories at all, which means that as such, there is
no need for EFL/ESL teachers/learners to waste their time on explaining/trying to
understand them.

If I am not going to be regarded as complete anathema right away, I will only be happy to discuss these and all the rest of the issues with you.

Of course languages are not designed, but terminologies are.
The terminology in “the use of verbal structures in English has indeed evolved into something arguably too finely nuanced or just plain quirky.” Now, was it not a case of intelligent design?
I think Warsaw Will’s “A Brief History of Tense” http://random-idea-english.blogspot.com/2014/01/a-brief-history-of-tense.html (Simply great! Many thanks for this piece of work!) provides more than sufficiently convincing proof of that.
Or, if we need more:
“How many tenses are there in English? This is a simple question, to which, however, almost every linguist gives a different answer.“
“…2500 years of research have not led to any precise or universally acknowledged definition of the category 'tense'…”
http://www.glottopedia.org/index.php/Tenses_of_English#What_is_a_Tense.3F

It is clear that nothing is clear to anyone, and hasn’t been for 2500 (!!!) years.
Just think: How does a foreign teacher/learner feel about and what is he/she supposed to do with all this?

In reality, the number one problem of the English tenses lies in the terminology we use trying to describe/explain what tenses are, what they mean and what they do.
“The focus of linguistics is not English, but all the languages of the world.” (“How Many Tenses in English?” by Maeve Maddox, http://www.dailywritingtips.com/how-many-tenses-in-english/).
“As far as I'm concerned, this is just EFL writers being trendy, like the fashion for calling Phrasal verbs Multiword verbs. It doesn't help the students one jot, unless they're going to go on to study linguistics at an English-speaking university…
The needs of linguistics and language teaching are very different: linguistics is mainly to do with analysis, not teaching.” http://random-idea-english.blogspot.com/2014/01/a-brief-history-of-tense.html
Like the authors of the articles, I am strongly of the opinion that for teaching, we should only use simple/major grammatical terms letting the rest of them go to the labyrinths of linguistics. The problems will disappear all by themselves.