I just finished reading a Coinage article about suspect coins that left the mint. They spoke at length on the 1804 restrikes and the 1913 strikes. After reading that article, I agree that they may not deserve to be called coins. What makes them coins? They were struck at the mint with mint equipment by mint employees but they were not fairly disbursed.
I've always thought the 1804 dollars were far, far, far overrated. If they are struck 50+ years after the original coins were minted, that's not a coin to me, that's a copy.
The 1913 nickels I will call legit but i'm biased. I love that series and the myth of those coins is what drew me to collecting.
As for why we pay for these objects, that's all in the eye of the beholder. I wouldn't pay for the 1804 dollars because, to me, they are not coins. I would pay great sums for the 1894-S dimes or the 1964 dollars since those are two series I love. I pay what I think the coin is worth to me. I start series and work on them because I like the coins, the challenge, the designs, etc. I'm willing to pay the market price if I think it's worth it.