@George Carter
The world always needs more people with an innovating mind as yours and the great thing about people like you is that they know that most of the concepts they think about won't work but they keep searching until they find the right one. I think that this is one to pass for the following reasons on top of those already mentioned:
1. You're trying to mix lifestyle and investments, which are two different animals that don't necessarily go along with each other. It makes much more financial sense for investors to invest where they make the highest return and then pay for a stay in the short-term rentals of their choice when they want to travel. That's even better from a lifestyle point of view because you can go to different properties every time, have a much bigger choice of properties and stay in properties that are much more luxurious than those you would be a co-owner in rather than be limited to the ones you own.
Mind you, many people buy lifestyle properties overseas but that is because they want to stay regularly in their own house the decoration of which fits their taste and they feel at home, can leave their stuff there,... You won't have that benefit of ownership if you share ownership with others.
2. If I want to invest my money overseas with people in countries A, B, C and D, I'll want to invest in A, I'll give my money the best qualified person who knows the ins and outs of that market. Same with B, C and D. I won't give my money to somebody to invest in A, B, C and D who's a jack of all trades and master of none (this is obviously no personal judgement of you). Investing internationally is a huge opportunity, especially now that the US model based on cheap debt is kinda broken, but investors need to know what they're doing or they need to work with people who know what they're doing. And investing in country A can be very different than investing in country B.
To conclude, even though international investing is my passion and I've been doing it for years, I think such a concept is very unappealing. It sounds a bit nice and glamorous at first but, when you start to dig a bit deeper, it makes no sense to me. The only thing you'd get out of it is earn some bragging rights so that you can say you own property in A, B, C and D. I don't know about you but I rather make money than brag. This, notwithstanding all the other issues that have been mentioned.
Being innovative and a trailblazer is great but it has to make sense, because trying to reinvent the wheel is often the thing not to do. This isn't rocket science and could have been done many years ago. If it was a workable concept, wouldn't it have been done already?