You are partly correct. A small percentage of coops cannot ever be rented out. These tend to be found, but not exclusively on the Upper East Side here in Manhattan. The typical coop does allow very limited subletting. The usual policy (but not standard) is that you must own and reside in your unit for at least 1-2 years. After owning and residing at least part time for a couple of years, some coops will allow you to sublet for a 1-3 year period, typically. Some coops are more liberal and have an unlimited sublet policy after owning for a period of time. Purchasing with cash does not change this fact. There are also "condops" which will condo subletting rules (very liberal, few restrictions at all) but the purchasing procedure of a coop. You should also be aware that with a coop sublet, your prospective will be required to interview with the coop board and be approved as a tenant based on his/her financial outlook, personality traits, and just about anything the coop decides is important to their coop community. Some coops are very particular and a pain in the neck with subtenant selection.
As a foreign investor you should also know that almost 100% (but not all) coops will require their owners to have an established credit history in the United States. They will also require you to have between 3 months and 3 years of mortgage (if you carry one) and monthly maintenance fees in liquid cash in US banks after closing.
Purchasing a condo as an investment is the BEST bet in Manhattan. For starters, a condo is real property, as in you own your apartment. A coop is owning shares in a corporation. You don't have the same strict financial requirements (especially if paying cash) and your subletting policies are VERY liberal and allow for many creative and cash generating possibilities for the unit when you are not occupying it yourself. (i.e. you may sublet immediately!) If you click on my profile icon and read my answered messages, you can see some of the options that I have presented my foreign investors with and that they used to maximise financial gains.
If you have more questions or would like to speak at length as this is very complex process your first time, my business partner and I would be happy to help.