I wasn't the person who recommended a rescue greyhound, though I do have one, so I feel the need to put in a good word for them! It's true that the ex racers have only ever lived in kennels, but most transition to home life relatively easily with a little time and patience, certainly much less effort than would be required with a puppy of another breed! Similarly while they haven't been formally toilet trained, most are 'clean' in kennels and are able to transfer this to the home environment fairly easily and instinctively - ours was toilet trained essentially from day 1 without any active training, we had at most 1 or 2 (wee) accidents in the first week while he was settling in and then he was perfect.
It's true that it is difficult to train them to a level of recall where you can feel comfortable letting them off lead in e.g. busy public parks, but then again most of the 'family-friendly' dog breeds we see in the park are not trained to that level either, as evidenced by the fact that they are in the face of my on-lead dog almost daily and can't be called off by their owners. When greyhounds do have chance to run off-lead in a secure space, it is a truly amazing sight!
Prey drive - only a very small minority of greyhounds are prey-driven to the extent that they require muzzling, and a rescue can easily match a lower-drive dog with first-time owners or a family with young kids. My dog totally ignores cats and squirrels if he is on lead. Meanwhile I frequently see squirrels, ducks etc. being terrorised by off-lead spaniels...
Playfulness - they are not typically 'playful' in the sense that they will play tug of war or chase a ball (though the odd ones will), but they are dopey and goofy and silly and provide hours of entertainment (mine loves throwing his soft toys around). They have so many quirky mannerisms that are fun to observe. They will never be able/willing to go for miles like a border collie, but they can definitely be acclimatised to longer and more intrepid walks if that is what the owners want to do. Equally, if the weather is crap and they only get toilet breaks or a quick round the block for a day or two, they are totally fine with that and will happily sleep the day away. Ours walked beautifully on a loose lead from day one. I don't know how that isn't the perfect family dog, or at least close to it!
Sorry as this post isn't really directed at you or even at the OP, but I feel duty-bound to counter the misconceptions in case anyone else is reading and is put off!