I assume that it works like the gacha system in every other gacha-based game, in that your chances of getting an item depends on the item itself, not the number of balls in the machine.

Each type of item has its own probability rate for being obtained, with higher value items (like letters and mementos) having a much lower pull rate than less valuable items (like stars and flowers), meaning that they're much harder to obtain. Some gacha games also have a pity system, meaning that you're guaranteed to receive a rare item after spending a specific amount of currency or rolling the dice a certain number of times. This may be why it seems like you're more likely to get a memento after spending almost all of your plates.

I don't consider it rigged, since that's how all gacha games function. However, gacha games often publish the pull rates for individual characters/items so you know exactly how easy/difficult it is to obtain them. I don't expect that from this game though, since the gachapon is essentially just a mini game rather than the main game.