I've been playing piano for 30 years, have a degree in piano performance and have played in many competitions.
There are many type of competitions. There's the Chopin International Competition where only Chopin must be played. Also the Liszt, and others. There are also Bach competitions, which don't require large hands.
Then there's the Rubinstein and Van Cliburn where you have to adhere to a set of music. Like one sonata from the romantic period etc. Many different competitions and opportunities to play many different pieces with many different requirements.

I've also watched tons of competitions and know tons of pianists.
I think a smaller keyboard would make it easier for people with smaller hands, but a lot of these "big chords" that composers write, are not playable by anyone, and were never meant to be played as one chord. Rachmaninoff has chords that regularly stretch 3 octaves. No one has hands like that.

I've also met a ton of concert pianists (name dropping here, but for a reason).
I never met Alicia de Larrocha but she was 4 ft 9 and she played tons of very large pieces like Liszt.
I've have met Helen Grimaud, Evgeny Kissin, Lang Lang, Jean-Yves Thibaudet, Andre Watts, Daniel Barenboim and a bunch of others. I know Lang Lang has largish hands, but the rest have hands very similar to mine, and I'm perfectly average.
Except Andre Watts, who had super thick hands and regular length fingers. I remember that Helen Grimaud was small, but had the strongest handshake. All of them can play all the great pieces.
Also in conservatories you'll often see a these kids that can play crazy pieces.

Do large hands help. Yes. But like anything, we adapt, and you learn your own technique. I don't think it's the same as having long legs in running. The biggest difference would be for people who can't reach an octave comfortably. After that, 10th, 11th, 12th, 13th. You're in the same camp as the rest of us. Some I've seen can reach a 10th comfortably. But I can play a 10th, if I stretch, but I don't because that stretching will injure my hand over time. I play it as a quick roll.

I've also analyzed the technique of many of the greats in slow motion. Go watch Evgeny Kissin at 1/4 speed on YouTube. He isn't using his hand size to make the difference, he's just very efficient in his movement. He's the same height as me, but maybe with his fro slightly taller.

Look up Dorothy Taubman, and Edna Golandsky. I studied the Taubman techinique for years. What she did is (in the 70's) rented slow motion cameras and recorded great pianists and not so great pianists, and she realized, it wasn't so much the hand size but they way they used their hands. Great pianists had a natural rotations between notes. They were very naturally efficient. They made large jumps by moving quickly to the notes and then playing vs flying blindly. They move their hands and fingers to play notes vs just playing notes with finger movt. Many large discoveries, She turned it into a whole technique which Edna Golandsky later took over and made the technique her own. I had a lesson with Edna, and it was life changing. It was the realization that it doesn't matter what your hand size is but how you use your fingers, hands, arms and body together. I've been playing wrong for years. Because no one teaches you how to move. There's the "Russian technique" which will give you tendonitis. The movement hasn't been studied as much as it should be.
I learned 10 or so major concepts and swore off all other music for the summer and retrained everything. I studied that technique for a few years, and now I am able to play the Liszt's and Rachmaninoff's, given enough practice. It was that I was playing incorrectly. I didn't know how to move my body. It's an efficiency thing, that some discover naturally.

My biggest handicap is my learning speed. I personally think that the real thing that makes the greats great is their ability to naturally sight read. Think about it. If you can sight read very quickly you have solved all other problems with movement. Watch Daniel Barenboim and Vladimir Ashkenazy on YouTube sight reading together as 20 somethings. This is something they are very naturally talented at.

Anyways, key size is only one tiny piece that I think will have diminishing returns.