How sure I will get the 240 transit visa?

Your itinerary is definitely eligible and there is no reason that you, as a US citizen, are not granted a permit. The only exceptions are the persons with some previous breaching the immigration and other Chinese rules and those who do not have an eligible itinerary. Also, it is important to use a correct terminology because it is not 240 transit visa, but 240 visa free transit, so there is no visa of any kind at all. Also, when you land in Beijing the very first thing you will do is to approach 24/240 hour visa free transit counter where you will apply for a temporary entry permit by presenting your flight tickets and this application is free of any charge. It is advisable to have an accommodation confirmation for at least the first overnight in mainland China. Just be relaxed and enjoy your travel.