Having rescued a few Labrador and now adopted a racing greyhound, I've found that dealing with dogs that are fearful takes patience. Luckily Hoffa wasn't abused but did come with some habits that required patience. For example, she evidently had to compete for food so for the first few months after her arrival, she would gobble her food and also growl if Tilly or Stella came near her. Now, she has gotten to be a picky eater and takes her sweet time finishing up her breakfast and dinner.
I don't know how common it is for racing greyhounds to be abused. Certainly there can be kennel workers who do not handle the dogs properly. Hopefully, kennel and dog owners immediately fire anyone caught abusing a racing dog.
These dogs are an investment for the owner and most owners probably want to make sure that they are cared for and that they like to run. From what I've read, abused dogs tend to not run. Or at least, they do not run well.
Having a greyhound has been a different experience from having Labradors. The Labradors don't suffer much anxiety about anything. And they always like to eat. Greyhounds, like all sighthounds, are keen to movement. Sudden movement that catches them unware will most certainly trigger one of two responses: fight or flight. Greyhounds by nature will usually choose to flee an uncertain situation.
And from what I've read, an abused dog really shouldn't be treated any differently than a un-abused dog. It's important to not force the dog into a situation that stresses it out. But it's also important to introduce the dog to different situations. I've found that it isn't a good idea to coddle dogs too much or to love on them when they are fearful. It is human nature to want to reassure the dog but to a dog, such reassurance can be viewed as a reward. It's best to remain calm and indicate that the stick or the car or stores aren't as frightening as the dog thinks they may be. If possible, make these activities fun for the dog and give praise when the dog doesn't react in a negative for fearful manner.
Acclimation for a rehomed dog can take as long as a year. I've gradually introduced Hoffa to all kinds of activities such as riding in the car, going to the beach and to doggy park. She now loves to go for rides, is the greeter at the dog park, and enjoys the beach. Patience wins out in the long run.