But after I attempted a test fit, the steel rod would not slide in. And it's not like it was close, there was no way the steel rod was going to fit. In fact, it felt like it was almost too large to even be an interference fit.

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I wanted to make a hole that would allow a nice slip fit of a 3/32" (0.09375") steel rod. Though it was actually slightly undersized by up to 5 tenths. I used a #43 drill bit (0.0890") to start, and followed up with a 0.0938 reamer. I performed the task on the lathe to make sure everything was square, and the entire process went smoothly.

As I understand it, you can use reaming to create a hole with tight tolerance and a smooth wall. If you want to ream a hole with X diameter, you first drill a slightly undersized hole (~98% of the final diameter), and then finish with a reamer of the actual diameter.