I guess I'd agree with skipping The cage, because it's embedded in The Menagerie as a flashback. But I would recommend starting with Where No Man..., because, ya know, pilot episode. After that, watching in production order is fine.
As it turns out, broadcast order is pretty random. It's fascinating reading the book "These are the Voyages" and seeing how much trouble they had getting episodes finished in the beginning! Corbomite Maneuver was supposed to be first, but the effects weren't ready, Charlie-X was almost ready but the network thought it was too slow, Man Trap was ready, so they decided to show it as the premiere, even though it was the fifth ep filmed... Then they showed Where No Man... after a few regular epiisodes because there was no new ep ready yet, and everybody wondered why everything looked so different and there was a different doctor... A big mess, it was.
Anyhoo, yeah, I think production order is the best way.