Breeding is a factor with pretty much anything when it comes to dogs. Hypothetically, you'd want a dog fast enough to at least sort of keep up with the deer, sturdy enough that Bambi isn't going to beat him up too badly if he gets too close, and smart enough to know that Bambi can be pretty mean if he has to be. This, of course, is on top of having enough of a prey drive to actually want to chase after Bambi, and trainable enough to learn how to do it so Bambi doesn't kick the crap out of him.

Putting all that into the Dog-O-Matic(TM) Computer....

... WORKING...

... beep boop Beep beep ker-chunk boop boop

... DING!

THE MOST SUITABLE DOG FOR YOUR REQUIREMENTS IS: the Scottish Deerhound[0].

====================

Of course, there are other dogs that would do just as well, like the Norwegian Elkhound[1], the Karelian Bear Dog[2,3],
the Bluetick Coonhound[4, 5], or, even that well-known gentle giant, the Great Dane [6]. Believe it or not, Great Danes can run really, really fast. Their top speed is somewhere in the neighborhood of 30 mph[7]! Even Bambi might not be able to outrun that.

Whichever breed you choose, get your dog from a reputable breeder of working line dogs. Working dogs, show dogs, and pet dogs all tend to have different temperaments, and you'll have an easier time if you pick a dog from a line that's suited to his role. Frequently, the different lines will even have different physical characteristics[8, 9].

Not only that, but you'll want to make sure you pick a good breeder, and choose the dogs that will be most likely able to succeed at running down Bambi. For hunting dog breeds, at minimum, this means the dog's parents should have passed breed standard hunting trials, or the equivalent. The ultimate would be a pup whose parents are both field champions, or champions in some other doggie sport[11]. You want a star canine athlete, not a bench warmer.

Anyway...

TL;DR: Yes, breeding matters. No, you wouldn't have to create an entirely new dog breed to chase Bambi around. Pick a good breeder. Choose a dog based on his parents. You're looking for an athlete, not a pet.

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[0]: https://www.akc.org/dog-breeds/scottish-deerhound/

[1]: https://www.akc.org/dog-breeds/norwegian-elkhound/

[2]: That sounds made up, doesn't it? Seriously, it sounds like something straight out of Star Trek.

[3]: https://www.akc.org/dog-breeds/karelian-bear-dog/

[4]: Watch out when talking about your Coonhound in polite company. These dogs live to chase and corner raccoons ("coons") up in trees, barking at them from the base of the tree while the hunter shoots at them.

Why be careful? Unfortunately, "coon" is an extremely offensive racial slur against Black people in the US. You really want to avoid that type of association with dogs, Black people, and trees, if you know what I mean.

See https://www.ferris.edu/jimcrow/coon/

[5]: https://www.akc.org/dog-breeds/bluetick-coonhound/

[6]: https://www.akc.org/dog-breeds/great-dane/

[7]: https://highlandcanine.com/the-fastest-dog-breeds-in-the-wor...

[8]: https://www.allshepherd.com/working-lines-german-shepherd-vs...

[9]: https://www.allshepherd.com/german-shepherd-sloping-back-vs-...

[10]: https://www.akcchf.org/canine-health/canine-athlete-performa...

[11]: https://en.wikipedia.org/wiki/Schutzhund