Maybe I'm clueless, but I've been on two mission teams and neither one looked like anything I'm reading about in this thread. In both cases, I spent my own money--plane ticket, food, place to stay, etc. No church gave me any money.
The first trip was for two months in South Korea. I stayed at some missionaries' apartment while they were in the US (and the apartment was ten times cleaner when I left than when I got there), and I taught at their Christian school because the teachers were in the US, and I played the piano at church because the musician was in the US. I bought and cooked my own food, visited students, passed out tracks, etc.
The second trip was for two weeks in Cameroon. We went to visit some missionaries who LIVE there. Not "now I'm on furlough, now I'm not", but missionaries who have a house they built on a mountain and they never leave. A small group of people (20-40 year olds, not teens) went to encourage the missionaries. One person in our group had taught at a seminary in Cameroon and was returning to speak at the graduation ceremony that was going on. While we were there, we held Bible studies, took medical clinics to the jungle, held a Bible Conference, helped build cabins for a Christian camp, etc. When I left, the missionary's oldest daughter (my age) told me that these were some of the most encouraging weeks of her life.
I guess as long as I call both of these my "vacations" that I paid for, then I'm okay. But if I call them "mission teams" then it was a waste of time and money?
Also, the church I attend now sends our pastor to visit our missionaries on a regular basis. He is able to come back to report what has been going on, how we can better pray for the missionaries, etc., and our missionaries tell us that one of the biggest encouragements they have is knowing that our church cares so much about them and their ministry.