The lee valley thing is just a reaming bit, the instructions show using a standard pencil sharpener to sharpen the dowels. So just use a pencil sharpener if you want to try it that way. You could also probably just take the tiniest bit off of the dowel end and drive it in, even oak dowels at 1/8" will flex a little. Like just rub the end against a sanding block then tap it in. Like you were planning I think, I would buy a bunch of dowels then cut them into 10" or so, drive them all in, flush trim them all then resharpen the end.
Hmm with 1/8" you can probably use snips or something to trim close to flush, and that would hopefully create enough bevel or whatever to the end.
Don't understand the patchmaker or routing holes suggestions at all. You have a bunch of (assuming) properly sized holes already, and you can buy 1/8" dowels... I would just go at it and work out the fastest workflow system. I bet the floor will look awesome though. End grain or some kind of subtle contrast seems fine given that no matter how carefully you select the wood for patches, there is no way that 10,000 will be invisible.
