Look at the elder on the left's face. This conversation is not going well.
I remember hearing that. I remember saying that. And I remember the day I thought about it and realized why it didn't add up.
Now, I didn't come to this conclusion until after my mission, but I think that this realization helped me to not only realize a missionary's role better, but to better empathize with those people who so often slammed the door in my face.
But this leaves out a lot of factors—namely, faith and testimony. If this model were an accurate description of missionary work, it would still find a way to evoke these principles.
I propose instead the following adaptation.
For the purposes of this, we'll still assume that the world is plunged into darkness. It isn't, however, a pure darkness, but rather a sort of dark twilight. People go about their daily lives, some either growing accustomed to the dimness or just never noticing it, some feeling almost subconsciously stressed out or anxious because of it, and others feeling downright driven mad by of it. One day, along come some nice men or women talking about light. Depending on your personal feelings about the twilight, you may be immediately inclined to throw them out, listen to them, or something in between. In any case, your response makes perfect sense to you because of one crucial piece of this parable...
The men or women are not carrying light.
They do everything regarding light but carry it; they talk about light, preach about it, testify of it, read about it. They tell you that they know a way to get light, and that they represent a group of people who meet weekly to further discuss this eventual method of receiving light. As you meet with first these young visitors and eventually the weekly group, you may well become more and more open to believing that, yes, doing the things they suggest will in fact let you one day enjoy light.
It will probably come as no surprise that I prefer my revised model to the more common one. This model relies not on proof, but on faith (and I'm a fan of faith). It shows the importance of testimony—whether bearing or of receiving one. It underscores the (potential) value of having a congregation. And, perhaps most importantly, it gives insight as to why a person may reject the missionaries.
Now, this isn't a perfect model either; there are any number of potential problems whenever one tries to simplify something as complicated as proselytizing and altering people's core beliefs. As such, I welcome any comments you may have regarding this. I feel, though, that this is a more correct and more complete vision of missionary work than the previous paradigm, and I wish I'd thought this way while I was a missionary.