I never claimed that it is
inevitable that they'll sustain their current growth rate for the next 2-3 centuries, only that it is
entirely possible. And I don't see why it wouldn't be. If you look at ultra-Orthodox Jews in New York/New Jersey, I don't see how they'd hit any natural barriers to their continued growth until they are many times larger than they are now, at which point they'd be a serious challenger for becoming the population majority.
The main ways it might not happen would be if either (1) they (gradually or suddenly) abandon their current culture for a less fertile one, (2) mainstream society persecutes them sufficiently. Both are entirely possible, but neither is anywhere near certain.
The situation for the (Old Order) Amish is more difficult, since – unlike ultra-Orthodox Jews – they'll run out of enough land to sustain their agrarian lifestyle, and will have to transition to a more urban one. While the more urban lifestyle of ultra-Orthodox Jews demonstrates it is possible for insular high fertility religious minorities to exist in an urban setting, there is a real risk that they might lose their fertility and/or their insularity in the process.
Also, people often bring up the problem Israel has with many ultra-Orthodox Jews not working and relying on government subsidies to live. That is much less of a problem in the US than it is in Israel, so the sustainability of that lifestyle is less of a barrier to future growth in the US than it is in Israel. Furthermore, the fact they manage to grow so much in the US without doing that, means being forced to stop doing that isn't necessarily going to stop their growth in Israel either.