I've read that and it doesn't really answer those questions. How can you measure the core's neutron multiplication rate if you're not exactly controlling the distance? Isn't the measurement going to be all over the place?

In a demonstration, not an experiment, it’s sufficient to have the Geiger counter go clicky at different rates while the demonstrator plays the sphere like a theremin.

The point was to show it to people, not to collect data.

It was a boy's club with unlimited funding working on things that were never attempted before, a lot of things weren't exactly done by the books, even their original "safe" protocol would seem completely insane by modern standards. As long as it went boom in the end and they kept it secret I doubt they had many rules

Imagine if you'd invented the world's first modern sink, in a world that had never seen a faucet or a plughole before. And you're training some new guys on the details of what you're working on.

Sure, some of that training is going to involve blackboard calculations and careful measurements.

But you're also probably going to demonstrate a sink to them and say "As you can see, when we turn this knob more hot water is added to the mix. Note how, after I put the plug in the plughole, the water level starts rising."

The purpose of the demo isn't to precisely measure the depth of the water or the temperature at the faucet or the angle the tap is turned to. It's just to let them see the thing in practice, so as they study it in theory they know what to imagine and how the model maps onto the real world.