First I must answer your main question.
Does the teacher advertise to students what the things are they can
No. All teachers of all levels don't need to do it.
Does teaching help students to learn what they want to learn?
Is just giving some information teaching?
No. Teaching will not take place without reception.
Should there be a goal behind why some one is teaching something?
Yes. No one does anything without a purpose.
From your question, I understand that you are trying to find the role of teacher or teaching in the present scenario.
Though the word you used here is 'teacher', the apt word for the person who does the purpose you meant is 'educator'. And since he is the person who renders education, he is supposed to be a role-model of good qualities also. The word -- TEACHER is now being used as an umbrella-word. A linguist can easily interpret this word as a mnemonic or acronym and enthrone himself as a great teacher and expound his latest theories on teachers and teachings. But since he also teaches something we call him a teacher. Let us put these things aside.
So the only thing you should find out is, whether the teacher is supposed to be just for teaching/training or for rendering education, and if so what is EDUCATION. But before discussing it let me mention a few things about teachers.
In some contexts, the word -- 'Guru' is used as a substitute for God. https://en.wikipedia.org/wiki/Guru . A teacher does not have to be 'a person who realized the Truth' like great gurus. But since he is with several good qualities and also he, without pride, upholds the Truth always, his words often appear to the students as a beacon that guides the goal and sometimes like a lightning that makes them alert and to work for the welfare of the society considering nature and their nature. The consequences of the carelessness in teaching/learning/education without giving importance to 'nature' can now be seen in the present world.
Of the three words in this famous quote: "A teacher is a friend, philosopher and guide', the most abused word nowadays is 'friend'. It is through the teacher's attitude/approach/behaviour the student understands that he is someone who is nobler than a friend. And this nobility in character is the force that motivates the student to rise himself to the level of his teacher. And thus the rise of the society also. Eventually the purification process that takes place through the teacher’s experiences helps his real development also.
Since the teacher is the social engineer (or if he is supposed to be the social engineer), like a philosopher he carefully makes use of each and every opportunity (no matter whether it is good or bad) that comes before him and makes it a tool for social development/construction and lives like a candle which burns itself giving light to others. (You should remember that each student, on one hand, is a separate entity but on the other hand, he is a part of the society.)
Learning can occur even without human presence; so is teaching. Anything can teach you something. You might have read this in the Bhagavatam: https://www.speakingtree.in/blog/twenty-four-gurus-from-the-srimad-bhagavatam . Also, I know you have no doubt in the proverb that ‘experience is the best teacher’. So all of this reinforces your notion that the process that happens is 'learning'. Actually, the teacher's role is in teaching-learning process; not only in teaching. And these things force us to examine the definitions of education.
You may verify this: https://www.acs.edu.au/info/education/trends-opinions/what-is-education.aspx
You might have heard this quote of Swami Vivekananda: “Education is the manifestation of the perfection already in man.” These uneditable words can come out only from a great teacher/genius who manifested the perfection already in him. Since what actually happens is unveiling, this quote implies the teacher's role/necessity in learning process. It is through the teacher this manifestation process is rectified. He encourages his students to pursue their goal and sometimes control their ego when it is needed; to prevent the birth of educated/clever devils. This is impossible even for AI, the controller of future generations.
This world is full of information. And a student (Anyone with
curiosity) is the one who wants to understand these things. Where is
the teacher in this picture?
I have already answered your main question. If you want to see your teacher 'in them', try to verify using this: If you found this function as fulfilled in any of the four options you gave, you can confirm that your teacher is 'there'. If you found him in more than one option, you can confirm that he is there also.
I believe that besides the interference of social reformers such teachings or teachers (as mentioned above) are now the essential thing to solve the unrest and anarchy in many countries.
I believe now you will be ready to fold your four-rib umbrella and look into the sky of reality.