Gralick
All organisations are collections of people. Yes but some organisations are employers who pay their employees to do certain things, or charities which collect money to do certain things. The church is not like these. It's complicated. (I think).
But it's advertising" is rather the point, is it not? Well, you could say that or you could say it is sneaky! Subliminal advertising. I reckon however churches advertise themselves they will appeal to some and not to others simply in the method they do it! 
Gralick, have you been a part of a church? Recently? You said ...often not self-evident to churchpeople that 'mission', or serving the community, is an end in itself which brings its own advantages. I think it is not true of all Christians or all churches that they do not see serving others as an end in itself, I think they do see it as an end in itself. But this may well be true of some.
You said Most businesses understand this with no problem and have personnel devoted to keeping their 'mission' going. But the church is not a business, it is a family, or a body, it does not just exist to promote itself. It has other functions. Mission is one big part but not the only part.
Re When you say people who are not Christians seem a whole lot happier for us to do some stuff, not things they see as 'mission', you sound resentful of the expectation that you would do nice things without ramming some kind of message down people's throats.
I don't ram anything down anyone's throat, ever. I am only slightly resentful of your suggestion that I might! So I am not sure how I could be resentful of whatever people may or may not expect of the church. I guess I am surprised that people would be happy to take what we have to give in some situations but not understand we have things we may wish to say. They do not need to listen to those things, but I am not sure why anyone would imagine all we want to do is serve all the time. Some people do serve A LOT, but they may still want to talk about God, or to God. EG when we have a free meal we might still say 'grace'.
If you really want to know it is not about either warm fuzzies or advertising our specific church al the time. Although in this instance, with people new to the area it is about advertising where we are, but more to the point, who we are, what we do, and that the regular people of the village are welcome to join us. Some people do feel they can't just turn up, they are not sure what to wear, what will be expected of them (if anything) what will be there for the kids etc.
So some of what we do is about plain old advertising, but it is more. It is our commission, to go and tell all people about Jesus, that he is God, that he died on a cross for us, that he loves us, that we can be friends with him, that we can experience eternal life. These are the messages about God. But there is also a message about people. You are loved, you are valuable, you are wanted. It is pretty hard to run up to people in the street and say all that. So we build relationships through friendships and through contact with people not in the church and outside of it.
I have quite a few non-Christian friends so they are obviously not put off by overt evangelism! I hope. 
Of course we think it is important to provide help, to do things, to care, and we do not only do this so that we can give our messages. But to fail to give our message, we feel, would be to let people down. You are only worth a cup of coffee, or a cake, or a hedge trim, not to be told about the God whom we know, and love.
My comments about what others feel, who are not part of the church, is that they seem happy with the cake and coffee and surprised sometimes, that we have anything more to say.
All I am saying, is do not be surprised. We are not a group who meet to serve coffee and biscuits, although we do, and it is valuable, and of course ANY who want coffee and biscuits and no message are welcome to it. But to suggest that if we want to go further than providing coffee and biscuits we are somehow being 'unkind', is to fail to understand all that we feel we have to offer.
I am not sure what you do hope you are wrong about! My guess is, that you are, because I am not at all pushy or unpleasant, in the flesh! Not am I upset but these discussions.
When we share our message, we are not overstepping the ... generous act... and it does not become ...an explicit quid pro quo or a purchase. It is an invitation, to know the God we know, more. Those who follow a different religion or none may well not be interested, and that is fine for them. But if they are questioning or searching, they may well be interested. We do not believe we can 'make' anyone into a Christian. And I am monumentally sorry that in the past, both near and distant past, some Christians acted in any other way than a loving way to those they sought to speak to.
We may well get the 'warm fuzzies' but that is not why we do it!
I enjoy these chats so hope I have not offended you Gralick, or anyone else who is reading.