1. The median transaction size will not necessarily be your transaction size. The cost of your transaction may change based on the inputs and outputs of your transaction. That's why fees are usually measured in satoshi/byte.
2. The fee of 67,800 satoshi is for your transaction to be included in the next block. That site also shows information of how long it may take with a variety of different possible fees.
I found this to be quite a good article on Bitcoin Core's fee estimation. Personally, I would trust that the estimation is accurate.
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Great URL btw, thank you for sharing