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I think you could make these spacers using a table saw with a glue line ripping blade, or a good sharp high tooth count carbide blade, but you will need machinist calipers, preferably dial or digital calipers, or similar to measure both the bit cut and the final width of the spacer, then you will probably need to make several sets of spacers, dialing in the saw fence until you get the exact spacer width needed. A bit of a challenge, but it might be fun to try. Using a router bit to make the cuts will always produce the same width cut, but it won't be exactly the width of the bit. Make a test cut in the same kind of wood and measure it to find the dimension needed. The same bit cutting different woods will produce slightly different width cuts in each kind of wood. Saw blades do this too.
Once you know the exact width that the router bit is cutting in that kind of wood, making spacers that are 2X the width of that router bit cut may take a few tries, but is doable. After a few test attempts at making a box, you will know how tight or loose the result is. If tight, lightly shave the width of the spacers using the table saw at the original setting, but with a piece of thin paper between the spacer and the fence as you pass the spacer past the blade. It should shave off the spacer an amount equal to the thickness of the paper. If the next box joint is still too tight, use two sheets of paper and shave the spacers again without moving the table saw fence.
When I'm doing precision cutting, I use spacers quite frequently like this, even though I have a DRO on my table saw fence. Unlocking the fence and moving it over 0.003" doesn't always work as well as just adding a 0.003" thick piece of paper between the work piece and the fence. The 0.003" paper spacer is way more reliable for this. I have favorite pieces of cardboard, playing cards, blank credit cards, and several kinds and thicknesses of paper in my box of shims. Each has a slightly different thickness, and when I want "just a little" shaved off of a work piece, I always reach for my box of shims for help. Just use a precision caliper or micrometer to measure a bunch of shim candidates and write their thickness measurement on them. Then, when you need one, it's there in the box waiting for you. I frequently cut off and use small pieces, but sometimes the whole piece is needed. It will depend on what you are doing.
You don't need fancy cutting tools, but you do need good measuring tools, plus some tricks and techniques for getting the cut accuracy that you sometimes need. Using shims when you can't move your fence accurately enough is a great way to get more cutting accuracy out of your existing cutting tools. Use them for your routers, table saws, drill presses, or wherever you need to sneak up on an exact measurement.