While there is nothing better than a cat finding his or her forever home, there are several reasons why cats that are trapped during TNR are not typically brought to shelters or rescues. One reason is that the cats returning to their original environment after neutering prevents other (yet to be neutered) cats moving into the territory-- thus it controls the cat population in that area, particularly over a long period of time.
Another reason is because most cats that are trapped are feral. Feral cats are unsocialized to humans and usually have lived their entire lives outside. While kittens under eight weeks old who were born to a feral mother can usually be socialized within days, socializing older ferals is a much bigger undertaking. The time, resources, and experience it takes to socialize a feral cat over eight weeks old is too much of a strain on already weary rescues and shelters. In addition, even the most experienced cat handler can fail to socialize a feral. Some cats are, to be frank, "born to be wild".
Feral cats brought to shelters are very likely to be euthanized. They are virtually unadoptable to traditional homes and often are not in ideal physical condition either. Living on the streets can cause skin problems, severe dental disease, and exposure to diseases such as FIV and Feline Leukemia (Felv).
All of that being said, there are exceptions. Usually kittens under 8 weeks old are socialized and adopted to homes. Occasionally, a missing pet cat can be caught during TNR. It's usually very obvious to veterinary staff and seasoned trappers when a trapped cat isn't feral. If the owners cannot be located, the stray cat would likely go up for adoption through a rescue or shelter. There are also some fosters and cat sanctuaries that deal primarily with feral cats. If they have the room and resources, they could step up to work with the feral cats and put the blood, sweat, and tears (quite literally) it takes to socialize a feral cat older than eight weeks.
I personally volunteer with an organization that operates a cat sanctuary with mostly feral or formerly feral cats. One of my own cats was a feral at the Riperian Preserve in Gilbert, Arizona. She was socialized as an adult cat, against all odds, and now lives happily inside. The socialization process took years and involved volunteers well trained in animal behavior.
As you can see, most shelters and rescues that are already overfilled with pets simply do not have the enormous resources to take in entire neighborhood cat colonies after trapping and neutering.