Hi Sam,
There may very well not be an easier/automated way to do this.
It might be possible, to create a Tool in Geometry Nodes, that just cuts through everything in ‘real time’, but that would be quite advanced (and far beyond my current capabilities) and more work, than doing this by hand. But then again, you could give away, or sell that Tool, to people that would like to be able laser cut everything.
I haven’t watched the whole Tutorial you linked to, so maybe there would be a slight speed-up possible for specific use-cases (like yours), but it really sounds like a mostly manual job to me.
Sometimes things that seem like they should be easy, are actually quite hard in computer graphics.