I would agree, Danes are out. The blood lines have been to heavily bred for confirmation showing.
The Irish Wolfhound is nearly as bad, though there are breeders focused on coursing strengths. Borzoi (Russian Wolfhound) basically the same.
The Scottish Deerhound has a much stronger following of breeders in the field (coursing) dog bloodlines. Be aware their intense prey drive means that small animals like cats, toy poodles (any dust mop dog really) and barnyard fowl tend to be fare game for chase.

I used to course coyote with an IWH and know for fact one IWH can hold it's own with one coyote and three greyhounds.

That makes me think of an alternative.
If you have a dog track local look into adopting two or more of the retired race dogs. Greyhounds make great family dogs. But then again, their prey drive and small animals,,,

"There are three kinds of men
Those who learn by reading
The few who learn by observing
The rest have to pee on the
electric fence and find out
for themselves."