Following the methods of the paper I already see a huge flaw.

They calculated energy per coin using hashpower from 2016-2018, ignoring all mining and coins from 2009-2016. Basically cherry picking their data and ignoring all the coins that were minted with lower hash power.

Surely the current cost of mining is what’s relevant when comparing the two today. Are we also supposed to go back and calculate the energy use for all the gold that was mined centuries or millennia ago?

It's different because the block reward is very biased at the beginning. By 2016-07-09, 75% of ALL bitcoin that will ever exist has already been mined. I doubt that's the same for gold reserves on earth. Furthermore, mining serves a dual purpose of also securing transactions, so for a fair comparison, you'd also need to factor in the costs of securing gold for storage/transport.