Does teaching Buddhism work?

What a beautiful and profound question. As a fellow teacher (Middle School) I share your wonder. Can we ever really learn from each other? Can we really ease another's path by sharing what we have learned so far?

There are some moments when I think I might have made a significant difference in another's life, but it was not solely my doing. I was fortunate enough to be the right person, at the right time, in the right mood, and with the right turn of phrase. That is all. Teachers live for such moments, but they are painfully rare and unpredictable. They can not be engineered.

The example of the Buddha shows that having a teacher is not essential for learning, and that it is the dedication of the seeker that matters most. In fact, I think there is no hope if this is not true, simply because if having a teacher were essential, then the first teacher would have already needed a teacher, ad absurdum.

As far as I can see, learning begins when the student is ready, and painful as it may be, would we really want it any other way?

It is called the "miracle of instruction" for a good reason; it is easy to impart information; it is much more difficult to actually teach:

‘And what is the miracle of instruction? Here, Kevaddha, a monk gives instruction as follows: “Consider in this way, don’t consider in that, direct your mind this way, not that way, give up that, gain this and persevere in it.” That, Kevaddha, is called the miracle of instruction.

‘Again, Kevaddha, a Tathāgata arises in the world, an Arahant, fully-enlightened Buddha, endowed with wisdom and conduct, Well-Farer, Knower of the worlds, incomparable Trainer of men to be tamed, Teacher of gods and humans, enlightened and blessed. He, having realised it by his own super-knowledge, proclaims this world with its devas, māras and Brahmās, its princes and people. He preaches the Dhamma which is lovely in its beginning, lovely in its middle, lovely in its ending, in the spirit and in the letter, and displays the fully-perfected and purified holy life. A disciple goes forth and practises the moralities (Sutta 2, verses 41 — 63). He guards the sense-doors and attains the four jhānas (Sutta 2, verses 64 — 82); he attains various insights (Sutta 2, verses 83-84); he realises the Four Noble Truths, the path and the cessation of the corruptions (Sutta 2, verses 85-97),236 and he knows: “...There is nothing further here.” That, Kevaddha, is called the miracle of instruction.

-- DN 11 (Walshe, Trans)

Still, I find this question baffling. Unless you think that Buddhists have gained nothing from the past 2500 years of following the Buddha's teachings, how could one think it not useful to teach?

Admittedly, some people (perhaps like yourself) are able to find the path to enlightenment themselves; a far greater number of people follow the wrong path simply because it agrees with their predisposition, mistaking it for the true path and mistaking a non-enlightened state for an enlightened one. Many such people, upon meeting and conversing with an enlightened Buddha or one of his enlightened disciples were able to correct their practice and attain the right path and final goal. That, in a nutshell, is what teaching is for.

At any rate, the Buddha himself seemed to think teaching was a good thing, and that discouraging those who have realized the truth from sharing it was a very bad thing:

‘In the same way, Lohicca, if anyone should say: “Suppose an ascetic or Brahmin were to discover some good doctrine and thought he ought not to declare it to anyone else, for what can one man do for another?” he would be a source of danger to those young men of good family who, following the Dhamma and discipline taught by the Tathāgata, attain to such excellent distinction as to realise the fruit of Stream-Entry, of Once-Returning, of Non-Returning, of Arahantship — and to all who ripen the seeds of a rebirth in the deva-world. Being a source of danger to them, he is uncompassionate, and his heart is grounded in hostility, and that constitutes wrong view, which leads to ... hell or an animal rebirth.

-- DN 12 (Walshe, Trans)

There are two side to this. The teaching works when there is a eager recipient ready to receive the Dhamma. (Ripened in perfections.) And the recipient practices the Dhamma to reap benefit.

Also Buddhism is a practical religion where you have to practice the 3 fold training to reach Nirvana. As you said the best is that you choose a meditation center and star the practice with lesser discussion.

As Buddha himself said (AN 3.22), there are three types of people:

1. Those who can figure it out themselves, without a teacher.
2. Those who can't get it, even after having received the teaching.
3. Those who can get it, but only with teacher's help.

It is for the sake of the third category, however small, that we teach.

To use your own analogy, if you were trying to learn how to implement an OO design pattern in JavaScript, would it be more expedient for you to try and figure it out for yourself or to learn it from someone who had already mastered the concept? Buddhism can be a very technical religion. There are some problems that will arise (and meditation alone is chock full of them) that you just won't be able to get past without someone there to guide you along. What's worse, sometimes you don't even know what questions you should even be asking! You might be doing something totally wrong, but without a teacher there to point it out, you'd never be the wiser. You likewise might master somethings, but not know where to turn your attention next.

I don't think the importance of a teacher can be overstated. One of the qualities of the Buddha is that he is an anuttaro purisadammasarathi - an unsurpassed teacher of persons to be tamed! That wouldn't be in the Buddhanussati Gatha if it wasn't important!