I've been self-employed for 20 years, and the idea that I would need to justify vacation is honestly a bit foreign. I assume that I'm going to be taking vacations, and if a customer has specific availability needs, then that's just something that needs to be negotiated. If a potential customer would only accept continuous work with no vacation, I imagine I'd pass. It's important to bill at a rate high enough that unpaid vacation is covered, and bench time between gigs is covered. (After all, employers budget for their employees to take vacation, so you should also budget similarly.)

When you bill by the hour, it's natural to start thinking of everything in terms of lost time and money. And I actually think it's reasonable to think about the cost of taking time off, but if this is keeping you from taking vacation and spending time with your family, then you are severely undervaluing your time off. You can think of time off as a sort of cost, but it's a very small cost compared to the value it provides.

I don't have experience with working for customers in different cultures and across international borders, so I'm afraid I don't have any insight there. If I was in that situation, I think it would still come down to negotiating the vacation that I need.

I also don't have much experience with juggling several (>2) projects at once. In my experience, people seldom need just a little software engineering. If someone wants some software engineering, they usually want a LOT of it and don't want to compete for your time with other projects. So I tend to work on one project at a time, take vacations, and when the gig comes to an end, I look forward to taking a few months off to study.

So I suppose if I was in your shoes, I would insist on vacation, and not worry too much if I lost a customer that wanted to burn me out. (I acknowledge that working in the software industry puts me in a rather privileged position where I can be picky.)