This is an old technique. The idea of convolving the impulse response of a system with an input to yield the output is the basis of LTI systems theory. And while convolution isn't the most computationally light operation in the world, computers have been up to the task of practically performing such operations on audio signals for at least 20 years.
There's been some more recent work about more efficiently and accurately measuring the impulse responses of a space, but given they're using balloon pops, they're not using that work.
I think the reason they used balloon pops was because that was all they were allowed to use: many heritage or archaeological sites can be nervous about researchers bringing in large loudspeakers and amplifiers, whereas a portable recorder and a bag of balloons can feel a little more harmless. Most acoustics researchers are aware of and use those new techniques whenever it's possible!
These days, some of the simulation algorithms are slowly trickling into game engines, too, with the push for realistic VR. Unfortunately, these still require some larger quality/performance tradeoffs because they have to share the processor with all the other work the engine has to perform. But it is still quite convincing. So with this tech, you actually get to walk around in a space and hear the reverb adjusting in real time, even when the geometry of the space changes.