Rescues vary considerably with the advice they give about greyhounds being off lead. But remember, that first and foremost greyhounds are dogs and like all dogs can be taught recall, and recall is an important exercise to train, train, train and keep working on.
We have four greyhounds, have done fostering, and currently have a mix of our own and fosters, and the key issue is prey drive, which varies massively from dog to dog. We are risk averse owners and take a different approach with each dog based on our knowledge of them and the location we are in. So for our forthcoming holiday in Pembrokeshire, all four will be off the lead on the beach because there are rarely cats or hares there, and we can see clearly for miles. In an unknown, squirrel infested park with multiple exits onto roads that would be madness.
You should also be aware that a greyhound/sighthound that runs and then gets injured may have their insurance claim denied if the insurance co. thinks you were negligent in letting them off in that location. There was a very high profile case recently where two sighthounds were off lead and bolted onto a road and were hit - the owners had to do a lot of appeals for fundraising to meet their monster vet bills.
But remember that in many ways because recall is an issue, most sighthound owners are in fact MORE responsible - we only let them off where it's safe to do so, and we work assiduously on recall and training. Go to any park and you'll see streams of non-pointies cheerfully ignoring their owners when they are being called - it's actually very dangerous potentially and often antisocial.
Also, greyhounds are not necessarily lazy - young ones especially enjoy and thrive on exercise, training and a varied programme. All of our current four are seniors and though they sleep a lot in the house, they love their walks, attend training classes (we're doing Rally), sighthound playdates and a wide variety of other social/exercise activities, including racing after lures, sighthound agility, greyhound walks etc. One used to love running with DH but his arthritis now means he can't do that, but a younger dog could certainly enjoy this, e.g. Park Runs or shorter Cani X. We also pay to use a dedicated greyhound field regularly.
In summary, you can certainly get low prey drive greyhounds but as with any dog this doesn't let you off the hook in terms of making a commitment to training and being responsible about where/when to go offlead.