Posts 954 and 956 on this thread might be useful to you. US passport holders are among the eligible countries, unless something has VERY recently changed.

There two different approaches.

There's the 15-day visa waiver, for those in a group tour. It seems Americans are explicitly eligible for this.

Then there's the visa-on-arrival. Americans have for years been banned from getting visas-on-arrival in China, and I've seen nothing suggesting Hainan is an exception to this rule.

For Americans at least, the difference between a visa-on-arrival, sometimes called a port-of-entry visa, and a visa waiver, like TWOV, is important.

Yes Americans, like Canadians, are eligible for the 15 days.
Koreans, Russians and Germans are allowed to stay 21 days, all others are 15.

I did my visa application by myself at the Houston Consulate and I got my 10 year visa within 4 days with regular processing. I was surprise on how quick it was.

Why were you surprised at how quick it was? Processing times are well documented.

I kept on reading that people should get their visas 2-3 months in advance. Obviously 1 month is sufficient.

At the risk of being snarky, if you had read the PRC consulate websites, you would have known that 1 day is sufficient.

IIRC not all consulates offer expedited processing of visas.

Yes, and the incremental cost is only $20.

Plus the expedited fee to your visa service agent, which can be around a hundred dollars.

1. Please reread post 1159, the "by myself" part
2. Which PRC consulates in the USA do not offer expedited processing?

Officially, Houston no longer offers Emergency/1-day service, probably the policy preference of the current Consul. Best possible is now 2-3 day Rush processing. With or without a visa agent. The thing is, Houston has one of the lightest loads for visa processing of any location in the USA, so their Standard processing is only 4 days.

Having said that, I've noted that on each of the many times I've applied there (using a visa agent and always Standard processing) that my package gets to the agent by FedEx on day 0, agent submits application on Day 1, the visa is issued on Day 2 or 3, agent picks up and sends back to me on Day 3 or 4, and I have back in my hands (2-day return) no later than Day 6.

So it may be that Houston just doesn't want to promise a same- or next-day turnaround even though they are capable operationally of doing it. On a more general note, for nearly 99% of China visa applicants in the USA, a little bit of advance planning should render moot, the need for anything under than Standard processing. Particularly now that 10 year visas are available in L, M, and Q varieties (the 3 most often used categories for frequent or infrequent visitors not requiring special paperwork)--there's no need to try to time an application so precisely before a departure to China.

Some people find it hard to plan a time when they're sure that they can be without a passport for the time that's needed to get a Chinese visa through a visa service agent without paying for faster processing by the visa service, the consulate, or both.

One special case that often seems to take longer: U.S. citizen applicants born in China who formerly held Chinese citizenship.

Maybe. But usually the time problem is in locating all the extra documentation needed as a condition of even submitting the application. A lot of people, especially those who came to the USA prior to the mid/late 1980's and haven't been back, simply don't have what's being asked for--and older records in China are either nonexistent or lost. And of course, Chinese citizens who were (are) not in favor with the gov't for some particular reason. Often those applications will get held up, then denied. But that's not the situation of the majority of Chinese visa applicants in the USA.