With regards to the 'paying for the teachers' - I've organised a few of these. You go through a specialist company - www.studyexperiences.co.uk are good - & part of the deal is you get x teacher places per y kids, depending on risk assessment. So higher ratio for orienteering than the theatre, say.
Generally they'll book you into hotels, shows etc, provide packed lunches, a coach, flights where applicable & a guide. You'll get a good price because you'll be in some out of town Travelodge on a block booking.
The company then costs it all up per kid. Part of the cost is, yes, the rooms, theatre seats, coach seats & so forth for the accompanying staff.
We obviously don't get paid for our additional time if the trip is taking place overnight/at the weekend/in the holidays. Generally I've ended up out of pocket because of my own increased childcare costs.
They can be quite fun OR they can be a bloody nightmare (last year's IYA expedition, I'm looking at you) but they are always incredibly tiring & you're on constant alert so quite stressful.
Definitely not a free holiday 