Dear Master Nelson,
I think there is a some truth to your theory, but not enough that I think you should move your family across the world. It's true that 84% of BYU's students come from only 10 states, but I believe that number to be somewhat proportional to the number of viable applicants produced by those states. There are obviously a lot of factors that go into determining who's a viable candidate and who isn't (SAT/ACT scores, GPA, Church service), but for the purposes of this discussion I've chosen to focus on the number of seminary students a state/country produces in relation to how many students from that place attend BYU.
Using BYU admission numbers and CES enrollment statistics from 2010 I've put together a chart so you can compare enrollment and seminary involvement by location. However, keep in mind that I am not in any way a statistician, so my use of the word "significant" is not to be interpreted to mean that something is "statistically significant." I'll tell you the conclusions I draw from the numbers, but they're just my opinions and speculations (any real statisticians in our readership will undoubtedly wince a lot while reading my analysis).
A large chunk of BYU's student population comes from Utah. This is not surprising, seeing as how Utah has more LDS members and seminary enrollees than most countries. However, you will notice that despite producing 22.58% of the world's seminary students, only 12.7% of Utah's seminary students go on to study at BYU. Now look at the stats for California, which makes up the next highest portion of BYU's student body. It produces 4.78% of the world's seminary students (compared to Utah's 22.58%), but 23.22% of their seminary students go on to study at BYU (as compared to Utah's 12.70%). This tells us that there are some states where the number of seminary students (i.e. viable candidates) is not proportional to the number of students admitted to BYU from that state. Utah, Idaho, Arizona, and Nevada (four of the five states you believe to have the worst odds) all have less than 15% of their enrolled seminary students going on to attend BYU.
Now, I'm not going to pretend that I know the reason why that's the case, because there are any number of factors that are not accounted for here. It could be that those states have high LDS populations, making attending in-state universities an attractive alternative to attending BYU, whereas students from states with small LDS populations might welcome the (to them) unique opportunity of living in a dominantly LDS environment. It could also be that candidates from those states do not stand out enough from their competitors in terms of extra-curriculars and Church service.
I initially wanted to put together something comparing admission rates for U.S. residents vs. international applicants, but BYU's International Admissions Office never got back to me despite repeated assurances that they would both call and email me. I got the impression that they were hesitant to release any numbers to me, probably for fear that I would unearth some sort of discrepancy proving that they are somehow prejudiced against Albanian applicants, which would cause all Hell to break loose. All I know is that when you tell someone you'll email them back you had darn well better do it (that's right Patricia Ware, I'm talking about you). Suffice it to say, the only stats I gleaned from BYU's site are that international students make up 6% of BYU's student body, with 1,944 enrolled in the Fall of 2010. At least 1,483 (or 76%) of those students are LDS.
BYU does list the number of international students attending BYU, but they group it by region, and as I don't know BYU's definition of "Europe" vs. "Eastern Europe," nor which countries fit into the (slightly racist-sounding) category "Far East," I was only able to crunch the numbers for "Mexico & Central America" and "South America," as those are more easily defined. As you can see, the percentage of international seminary students who go on to attend BYU is quite low, as in, less than 1% from the listed regions. You could take that to mean that your children would have less competition, or it could indicate the difficulty of being accepted to BYU as an international student.
Now, if you truly wanted to live as far away from Provo as possible, that would put you in the middle of the Indian Ocean, with the closest land mass being part of Antarctica. The last I checked, the Church doesn't have many (which is to say any) wards meeting there. There is a branch in Madagascar, which is next-closest geographically, and the country had 319 seminary students as of 2010. If you really want your children to be stand-out applicants, you'll move to Poland, which had zero seminary students last year, or you could consider Croatia, Turkey, or Curacao, which each had a grand total of one. In terms of low seminary enrollment in the U.S., Vermont is where it's at. No, your children will certainly never want for leadership opportunities, not when there are only 85 seminary students in the whole state. You will want to consider the enormous cost of travel that comes with shipping your kids across the country/world several times a year to visit, but hey, they may resent you enough for moving them to the middle of nowhere that you won't have to worry about it.
Hoping you find a good fit for you and your family,