The carbon coating over the traces on most controllers is just cheaper than plated ones, it also keeps the copper from corroding up as an exposed copper pad alone would do that.
The Resistance difference there is a couple hundred ohms at best on a carbon pad versus closer to 0 on a plated one. You can take a 10k Resistor and complete the circuit and the button will register, so that makes no real difference at all.
The problem is that 'finger' style type of pad used for the D-pad there.
What's happening is, when you press in a single direction, down for example, a tiny bit of the left or right pad can make contact with the board because those don't get pressed straight down like a button contact does, the D-pad pivots from the center. If it were in that half circle shape, then it could only touch the bottom half of the right or left pad and that will not make a connection between the 2 contacts for it. That finger type has a lot more area for the 2 connections to be made, and made on only the bottom half where the pad can hit, so it's just a bad contact design for a D-pad. It's fine for the buttons, where the pads hit straight down on them, but for the D-pad that's at a slight angle, they're awful.
These are a couple of pics of XBOX controllers, but they're good to illustrate the issue. The top picture is the older half circle style, and the red area is where the contact pad hits if pressed straight up, but it could also press slightly to the left or right. Slightly right is shown in this case, and it's only able to touch half of the contact pad, so no misfire can happen.

With the 'finger' style pads, that same little bit of contact that can hit the board on the right will cause a misfire as the 2 contacts for right are in that area, while on the half circle style it will do nothing.

Get the boards made with the ENIG finish, Electroless Nickle Immersion Gold, which is still gold plating, but it's not as expensive as some place that is only going to plate the exposed pads after the fact.
Also, not sure where you're getting them made at, but upload your design to OSHPark and see what it costs there to have them done. The go in groups of 3, and at $5 a square inch on a 2in x 5in board that's $50 for 3, or $17 per when all said and done, but it's cheaper in the long run to test out a design that way versus pulling the trigger on a bigger batch that isn't 100% proven yet.