First, there's a long history of people doubting this, goes back to 1974
Moon landing conspiracy theories - origins. But these tend to be non scientists mainly. Or with limited understanding of spaceflight and astronomy.

To answer your question, they don't prove it because it is already clear that NASA went to the Moon, if you look at the evidence. There is nothing extra for anyone to prove.

In detail:

The basic thing here is - that the Moon is further away than most realize. The spacecraft on the Moon are far too small to see with even the Hubble space telescope never mind Earth based telescopes. But they have been spotted by modern spacecraft in orbit around the Moon.

As for doubting if they went there well I think the rocks returned bear witness enough - and you can watch the astronauts actually pick up a rock on the Moon in the videos - and see it back here on the Earth, and it is the same shape / colour / matches the context of the place they picked it up on the Moon. For numerous samples for every mission.

And they have micro-meteorite damage throughout. Even nowadays, even with millions of dollars worth of modern equipment I don't think you could simulate a single lunar rock with all its microscopic detail of micrometeorite impacts throughout. So can't be Earth materials. And no way could they be lunar meteorites which were only discovered in Antarctica long after the landings.

And scientists continue to learn new things from those rocks, studying them to answer questions that hadn't even been raised back in the 1970s, and using machines that detect things that we didn't know to look for back then.

Apart from that - I think that nowadays many people forget how primitive our technology was back then. Things that anyone can do with your digital phone or laptop or tablet were way beyond the most advanced technology available to the superpowers of USA and Russia back in the 1970s. The "State of the art" back then was Kubrick's "2001 a Space Odyssey" which seemed advanced at the time and was hugely expensive with much of the material they shot ending on the cutting room floor - but looking back, many (not all) of the special effect sequences seem dated, especially its Moon sequences.

Do you find it realistic? Would it fool a modern audience into thinking this is something that actually happened on the Moon rather than in a studio set?

At the time it did, but we were naive back then, not used to modern special effects.

There is no way would a modern audience mistake the Moon sequences of 2001 for an actual video of a lunar mission. Just looks like people in spacesuits walking slowly in an obviously fake (to modern eyes) landscape, mostly shot with fixed viewpoints. Same would be true for any "faked footage" of that time, if that ever did happen - we just didn't have the ability to fake such things back then.

I've been told in a couple of comments that some modern viewers do find the 2001 Moon sequence realistic to the extent that they could believe it was shot on the Moon like the Apollo videos - if so - well - I suppose that is subjective. Some movie special effects are more convincing to some people than to others. However, try looking at it again. Here are some of the tell tale signs:

• They’re walking clearly in normal gravity. Though they walk slowly, notice how their body falls rapidly to the ground by a few cms at the end of each step - it isn’t the “floating” motion of the Apollo astronauts.
• Their spacesuits are slightly dimpled - obviously the pressure outside is roughly the same as the pressure inside.
• The floodlights show a hazy effect such as you get with damp air, not dust.
  
  Thanks to Michiel Lombaers for listing these inaccuracies in a comment

But to anyone keen on astronomy who has been following the discoveries about the Moon as they unfolded during and after the mission, all that is beside the point really. There is just no way it could be fake. When I first heard the hoax allegations a few years back, I didn't have a moment of hesitation or doubt, it was just obviously not right - and then when I read them in detail - they got so many things wrong that almost anyone with a fairly basic understanding of astronomy would pick up on right away.

Actually, I have more reason to believe that humans landed on the Moon than I have to believe that I was born where my birth certificate says I was born.

I can't prove where I was born. I've got a birth certificate, and my parents told me where I was born, got childhood photographs, even a video of the place they say I was born, in which I appear briefly as a small child. But all of those could be faked in principle :).

If you get me to prove that I was born where I think I was, I can't prove it absolutely to the standard that is required by the Moon hoax theorists. It would be pretty hard to anyway. But that doesn't give me the slightest moment of doubt that I was born there :).

But you can prove it for the Moon, seems to me, even to the standard required by the Moon hoax theorists - apart from any other reason - because of the impossibility of faking those rocks with their micrometeorite impact features and the evidence that they are indeed the rocks picked up by the astronauts on the Moon.

And we now have photographs of the landing sites from orbit too.

It's sort of a bit like asking airpline pilots why they haven't disproved the flat earth hypothesis :).

Lots of more material on this in answers to related questions, including for instance the recent lunar orbital photographs of the Apollo landing sites see:

Robert Walker's answer to What are the best arguments for debunking Moon Landing Hoax Conspiracy Theories?

and the other answers on this page:
Did Neil Armstrong really land on the moon?