there’s not a terribly easy answer here, a lot of this is practice.
having a good jig is obviously great, but some jigs require reliable precision in the making, so you’re not really solving the “how do I cut this reliable straight line?” question. even a lot of professional woodwork is relying on more precise tools like planes and chisels to really square stuff up, even after sawing stuff to rough size.
my take on the ishinomaki stuff is that almost none of it requires perfect micrometer square precision to each cut, but it does require consistent angle cuts since you want this stuff to sit stable and to be able to stack nicely.
if I was trying to do this with just hand tools, I would get my hands on a sharp japanese pull saw, a marking knife to set an initial guide, an angle protractor (digital or analog), and some light clamps (in addition to the tools to get it all screwed together nicely).
the only real critical tip for japanese saws is that you always need to remind yourself to saw a lot lighter than you think. focus on your wrist and pull action and angles and do not try to get the saw to engage with the wood. it’ll do that all on it’s own.
it’s good to watch the reflection in the saw blade (nice even angles means the saw is perpendicular to the wood stock). you can also fairly easily create a small jig (just a square piece of wood) into which you can embed magnets that’ll hold the saw face flush to it while letting you begin the cut perfectly square. getting a good start is huge, because the saw wants to keep following that path.
I think much of the stuff in that link is basically made out of 2x4 stock, so none of the cuts are very large, so long as your stock is actually square (it often isn’t). also, these are using pretty cheap wood, so if you mess it up, just start over again.
you’ll develop a feel with some practice and you’ll cover small mistakes with excessive sanding, which will make you want to get better on your initial cuts. it’s a virtuous cycle.
