(and we generally don't have to pay any conversion/transaction fee to make a purchase with a CC)
Actually we the consumer DO pay the tx fee. More specifically, the merchant pays it to the payment processor and their cost is added onto what we are paying for things - we just do not see it as a separate charge. The Merchants fee is generally anything from 2-6% of the transaction value. That is how/why some (local of course) merchants will give a discount for using cash vs credit/debit cards.
Actually, you're wrong. The merchants will usually not charge the transaction fee to the customers.