Well, she did stay indoors at nite, sure. But she had a small yard to run around in. But you have a dog park and go there twice a day and I can see that would be fine.
I can see where space could be an issue, but my point is the OP said they s/he had a park within close walking distance, and is home during the day for excursions to said park. I think an Italian Greyhound would be fine in such a situation.
I remember this thread, and all I say here what I said there.
I would never let my ex-racing Greyhounds off leash in an unfenced area. Never ever ever never. They cannot be reliably trained to an emergency recall.
With an Italian greyhound, whether you could do it or not would depend on their level of prey drive, but I’d be still very, very leery of letting them run off leash. All it takes is one time for a dog to run off into the path of a car.
Most places have leash laws anyway, so even if you had a breed that could reliably be trained, it should still be on leash while out in public unless in a designated area where they were allowed to be off-leash. It’s not only the law, but a polite thing to do where you are out in public. Not everyone likes dogs and having your dog obviously under your control can make other people feel a lot more comfortable about being nearby.
The advice I was giving was intended to help if the dog ever accidentally escaped. I agree with you fully that dogs should always be leashed while in public unless they’re under very controlled cicumstances (such as a fenced-in dog park where such things are permissable.)
This is good advice, which I practice often with my dog. He’s broken his collar once (the nylon snap kind) and has managed to break a retractable leash (two things we no longer use). The emergency call probably saved his life. All bets are off if he catches a scent and goes into that ear-splitting yelping mode though.
A little more detail on my experience with Italian Greyhounds, specifically Clifford, who my mother calls the “Cary Grant of dogs” 
I did say I let him off the leash to run home and that where I live makes it safe to do so. I’ll add that he is now 6 years old. I probably would not have done that 4 years ago, but I know him well enough now to know he just wants to run directly back to our door. And I also know he will respond to my calls if something comes up.
In terms of travel - he is great in the car (never tried to fly w/ him). If he isn’t allowed wherever I’m traveling, it is a LOT easier to find someone to dog-sit Clifford than it was for my previous dog that was a bigger, messier, noisier, hairy mutt.
My only regret regarding Clifford’s adoption is that I didn’t also adopt another IG that was available at the time from the woman that was fostering this breed (that I incidentally found on the italiangreyhounds.org site). I often wonder whatever happened to Scooby, Cliff’s little buddy. Having a pair would not have been a significant amount more or caretaking. Hell, I’d have a whole herd if I could afford it.
With my post I wasn’t referring to your advice, which I thought was excellent - it’s always a brilliant idea to train for an emergency recall. What I’m saying is that with some breeds, all bets are off, despite any amount of attempts to train otherwise. If I had a Greyhound who accidently got loose, I’ve got my methods to try and get them to come back to me but I sure can’t rely on anything to work, so I have to be extra super careful to not let myself get into that position in the first place and anyone who deliberately risks letting a breed like that off leash in an unfenced area is asking for a dead dog.
And there are all too many people who, unlike yourself, think it’s just hunky dory to let a dog wander around off leash everywhere. Dogs have been banned from a local university-owned park here because people refused to obey the signs telling them to keep their dogs leashed. These jerks have just made it harder, once again, to try to do something fun with your dog.
That’s how I took the advice. I have a little terrier that will chase anything alive at the drop of a hat. She’s small and if someone comes into my side yard lots of times they don’t see her and she’s gotten out. She will come when I call but I’m not 100% confident that if she’s chasing something she will and it worries me.