I'll check my drill bits when I get home, but the pilot hole size is a likely possibility. Although I purchased replacements from bolt-depot and I believe the recommended pilot hole sizes came with my order. I'm usually good at reading/following explicit instructions. Anyway, the bottom line is that if I move up to a 1/4" hardwood dowel (harder than birch) with a #2 size screw, and the pilot holes are drilled to the correct specification, I'm likely to be okay.
If I go with the machine screws, I'll have more peace of mind about the durability of the work, but I will need to use a weak-medium threadlocker, and I will have to combat the occasional tinge of guilt that I have modified a piece of history with a historically-inaccurate repair approach.
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