When we played Catan every single day for a year during the 1990s, we ran into the dice variance issue plenty of time that we considered creating a bag with 36 chits with the dice distribution, guaranteeing that all dice results would come up in the expected ratios. Or maybe 72 chits to remove the predictability when the bag starts to get empty, and return the chits once you got a complete set of 36 out.

But as you can see, the idea quickly became way too complicated and we never implemented it.

Removing trading sounds like a terrible idea. Being able to monopolise a resource just to trade it for what you really need, is a feature, not a bug. Some resources are rare, and you need to take that into account during placement.