An association with my daughters school takes the 16 year olds there on a month long trip around America for the summer. No bursaries are offered and most kids go on this tour and most have wealthy parents. If you could afford it would you allow your child on this? It would be a struggle for us to afford to pay and I don't want her to think that this.is what normal people can afford to do as many of her friends are multi,millionaires summer
Regardless of state or private school, if I could afford it without scrimping then I would pay for the dc to go. If I could manage say half then I'd offer the opportunity to them to earn the other half. Some of the schools here do the month long trips. Ds1 did and was sponsored by a local company, others raised money by raffles with prizes donated by companies or cake sales, car washes etc.
If I could afford it, and it wouldn't affect the rest of the family having a holiday, etc, then I would happily send my DC.....I wouldn't have an issue with them feeling entitled, etc. However, it's not the sort of trip where I would feel guilty about not being able to pay for if I couldn't afford it. It does sound like a bit of a holiday, rather than a trip to immerse yourself in the language, for example.
Save your money and when she's in university, have her get a J-1 visa to work for a summer in the US. You can give her some money at that point to get herself situated and she will have a far more educational time in the US.
DD1 and then DD2 each went on a school trip abroad that cost about $3,500 (separately, many years apart). They managed to fund half the cost. Even still, I don't think it was worth it except as an exercise in making and saving money (so quite useful but not as a foreign trip).
I'd never send my teenager on a holiday and especially not with school. Because even for an adult being somewhere unknown with others you're very easily influenced to do stupid stuff. Like a lot do on holiday. And in mine experience schools are very good in doing stupid stuff with kids when on camp or holiday. Teachers that drink and get drunk. Going in groups in unknown area and try to find your way over there. Not sufficient supervision over the teenagers.
I wouldn't go myself let alone send my kid with a school out on a holiday for a month!
If I could afford it as you asked in your OP, yes of course. I would make it very clear that this was difficult for us to do but that I wanted her to have the experience and expect her to save any pocket money or earnings towards it.
TBH I'm not a big fan of refusing things just to show a lesson ie as in this case not everyone can afford this holiday and that's what we are showing you by not letting you go even though we can afford it at a push. Ime most children realise exactly what their family can or afford and act and ask accordingly. I don't think we give them enough credit.