Singing in other languages, or any sincere attempt to communicate in someone else's mother tongue, tells the native speaker that you'll reach out and meet them more than halfway. That's certainly how it was taken when the chorus I sing in performed a verse of a song in Russian when performing in St. Petersburg, RU. Despite coaching by a native speaker, our pronunciation was tenuous at best, yet our effort was thunderously appreciated. Comparable efforts were similarly appreciated when the other chorus touring with us sang a Russian folk song in its original language.

And this is also how it's been taken here at home when I perform French-language (well, Québécois or "Fr'Ontarien", really) songs, despite the fact that I often mess up elisions and pronunciation of final e's among other things. When singing in a language not my own to an audience that won't or may not understand the words, I feel a strong responsibility as an entertainer to make doubly sure the emotional story behind the words is very clear.

Re: ««...we know that foreigners only speak foreign when we are there. As soon as we leave they return to speaking English...»»
I tried to bite my tongue on this one, but that didn't stop my fingers from tapping a reply in spite of myself: Huh? I find this remark thoughtless at best, but mainly presumptuous and perhaps xenophobic. I suppose that's because in my country, native-born people may speak any number of languages, including but not limited to English, French, a slew of First Nations tongues, as well as parents' tongues such as Urdu, Italian, Polish and on and on. Maybe the remark was meant to be funny, but obviously that boat sailed without me.

Re: ««...there is no univerally understood way of writing out sounds in a way that will approximate to the correct pronunciation...»»
Have you checked out the International Phonetic Alphabet, IPA for short? It's those unfamiliar letters & symbols found in dictionaries, for example upside down e (schwa). Usually it's studied by linguists, but learning a few letters can be invaluable for recording sounds that don't exist in your mother tongue.