@Edgar, I use the following informal explanations. I don’t know whether it’s widespread, because I devised them on my own.

1. Verbal communication is mostly used to express a message to people, to make them think that what you say is true. Consider messages like “[Treat me like] I’m out of here” or “[Считайте, что] я побежал [отсюда]". When you say something like this, you want others to understand that you’ve made a decision about your nearest actions, and you should be treated like you’re already doing it. The part in brackets just repeats the reasons why people talk. No wonder that it’s omitted in many languages, not only Russian.

2. If you add a bit of questioning or incentive intonation to the phrases described above, you’ll get exactly the message people express with “Let’s go shopping!": “[Пусть все считают, что мы] пошли по магазинам!". The “Let” ("Пусть") part merely got transferred into the omitted part of the phrase; that’s why the resultant “Пошли…” phrase seems weird.

Don, could you or your wife sometime explain why the past tense of verbs of motion is often used to indicate present or immediate future in Russian some time? I have never seen or heard a good explanation of how a native speaker uses this: Let’s go = poekhali or poshli or pobezhal. Thanks.

Don responds: I’ve never seen a formal explanation of it. In English, of course, we do something similar in the phrases “I’m gone” or “I’m out of here.” In those cases the speaker is mentally picturing his departure as already completed, even though it hasn’t happened yet. I suspect the same thing is happening in the Russian equivalents; that is, the speaker is mentally picturing his departure as a done deal, and so he expresses it in the past perfective.

If it’s a perfective verb, how come it has forms in the present tense?

Don responds: Oy! That was an embarrasing cut and paste error. The entry has been fixed. Thanks for bringing it to my attention!