It looks that you're right that the rules do not directly say this. However, page 4 says "play continues until, at the end of a round, a player reveals that they have enough gold bars to win". This implies cards are face down.
As we play tested the game we found that when players have certainty about who is closest to victory it makes the choices les fun as the game goes on.
If the bricks are all equal then the player with the most bricks cannot convince anyone else to work with them and in the end the game just goes until everyone gets one brick short of victory. Adding in a small variance in the values seems to solve that because, even though it's not easy, you can convince people that all four of your bricks are 3s.
As to the randomness/luck based question. There is some luck to the game but in the end the player who communicates the best and catches the others in their planning is who wins.
This would suggest that while gold cards are face down, you have to show the number of cards you have during gameplay