The U.S. had significant colonial possessions, and bases in a lot of other places besides. Panama and Puerto Rico, Guam and the Marianas and Saipan, The Philipines, Hawaii. Bases in Iran, other mideast countries, the U.K., Germany, Canada, Greenland...
Not quite as many bases as were necessary for, say, Britain in the coal-powered steamships era, but still significant.
The independence of a nuclear-powered carrier is exaggerated. Yes, her nuclear fuel can last for years. But the carrier operates as part of a task force; it has to, for protection against air and submarine attack. The task force as a whole needs regular resupply: fuel for the other ships in the task force, fuel for the aviation wing, spare parts and supplies for almost everything, food, a hospital ship if they're in combat operations and take casualties.
Bombers from the United States can only reach anywhere in the world through in-flight refueling, which requires bases elsewhere in the world with tanker airplanes and fuel storage to accomplish that refueling. Looks like we're back to wanting lots of bases all over the place.