A true lottery map is different from a king-of-the-hill type map. This map seems more like the latter. If you want to make it as 'lottery-ish' as possible, then I would say that the asymmetrical distributions (6, 10, 12, etc.) are not as 'fair' as a lottery should be, since some people will start without any neighbours competing for the next neutral. This gives those players an advantage. That's fine if the distribution is random, but if it's manual then there's some advantage to starting in those positions.
In principle, you could create a very similar map with the ability to start in symmetrical positions for those numbers of players, but you just need to make sure your outer ring has the right number of territories which are multiples of 2, 3, and 5.
So, you could have 30 outer territories, cutting down to a 6 ring by having groups of 5 outer territories fighting over a single territory in the next inner ring. Then cut that down to 2 by having groups of 3 fighting over 1 territory in the 'finals' ring, and then the last two survivors fighting over the last central 'winner' territory.
If you give up on multiples of 5, you'll have a better time of it, so you could have 24 or 36 outer territories. Then you could have distributions (from 36) for 2, 4, 6, 9, 12, 18, or 36 players, all being symmetrical.