Does die unfairness visibly influence real-world games?
I think all it's really possible to say here is that yes, it's a real potential problem. Dice can be unfair, and if they are, you can't really avoid it with rolling technique, and it will skew your game. 

I think it's fairly uncommon with commercial games, but I'm not sure how that's relevant to your homemade dice. For what it's worth, my copy of Small World has a wooden die that I eventually found to be slightly unfair (based on test rolls, it has a statistically significant bias), so if it can happen with commercially manufactured wooden dice, I wouldn't be optimistic about homemade ones.

For your actual problem, initial playtesting with custom dice, I would personally just use regular dice with a mapping from numbers to your custom dice faces. If that bothers you too much, or you're to the point where you really need to be prototyping the actual design, I'd put stickers on real dice (preferably blank - it's easier, and they're likely more fair), rather than wooden cubes, so you know they started out as fair as possible, and if the stickers are the same size and placed the same on all the faces, you shouldn't have anything to worry about.