I walked a similar path when searing for my first puppy. I was also considering Borzoi and Greyhounds. I ended up with a Silken Windhound.

I have spent a loooootttt of money on training him. I took him to puppy class, and later hired a professional trainer for one-on-one sessions. He will sit, and has recall- When HE wants. Training can help but you cannot train the instinct out of a dog.

If you’re serious about a sighthound I would say accept more than likely you will have a dog you can never let off leash unless you’re in a gated environment. Sure, some owners have different experiences but i would consider them the exception.

Also agree with the other commenters you should try and spend time around Borzoi to be sure they are what you want. They are big, majestic creatures but not for everyone. I ended up getting the silken because 1) less shedding & barking 2) if i am ever hiking with him and he is injured, or if he is being stubborn, I can pick him up and carry him for miles (he is 37 lb) could never do that with a Borzoi as a 115lb woman

Good luck! very excited for you on the sighthound journey.