I think you are a little unfair in your comparison of bitcoin and gold. Gold has advantages over bitcoin, but bitcoin also has advantages over gold. Storing gold is more expensive then storing bitcoin. Transacting gold is much more difficult than transacting bitcoin. One of the downsides of bitcoin is that it could go to zero for some reason. I think gold has less of a risk of that, but other investments have a significant risk of going to zero.
I am not sure I understand the negative sum game you mention. The first payment miners get is (I don't know the proper name) a bounty, which is equivalent to a gold miner obtaining gold. This can be seen as a built-in inflation, which not only gold but also any fiat currency has. The difference is there is no known limit on this inflation factor for gold or fiat currencies. For gold we assume some gargantuan deposit will not be found and that it will maintain its scarcity. I don't think anyone thinks there is an inflation limit for the US dollar or any other fiat currency. Bitcoin has a known amount limit. (As a disclosure, I own gold, bitcoin, and US dollars.)
The other part of what miners make is a transaction payment. That exists in every other payment system I assume, except I guess physically handing an object to another person, which covers a pretty small fraction of transactions.
AIUI, the negative-sum game breaks down like this: consider a black box around the entire bitcoin ecosystem. This black box is consuming $20B/year worth of real resources, which (given that you can't generally buy electricity or generation assets with bitcoin) means $20B/year is flowing out of it.
Now to be sure there are some customers of the bitcoin system, using bitcoin as an intermediary to move money across borders or purchase from certain online marketplaces, and those customers are paying some amount of money into the system to use bitcoin for this purpose (analogous to the 2% that VISA skims off the top when you use your card to buy something on Amazon), but the majority of the money flowing in to balance that $20B outflow is from investors buying in. Unless you think bitcoin has managed to capture something in the order of $20B/year in transaction fees from the likes of Western Union, AMEX and Paypal, what's mostly happening is money being funnelled from investors taken as a whole to the the supplier industries that bitcoin relies on.
You mentioned transaction fees, the other is "inflation" or mined coins. As mentioned above, this is the same as gold and many other things. I guess it is just a matter of definition if you want to say that is a negative sum game. It is not unique to bitcoin though.
It's important to realise that when I'm talking about transaction costs I'm not referring to the inbuilt transaction fees in the protocol - those are bitcoin-denominated and so are an internal transfer within the ecosystem (likewise for the block rewards).
I'm taking about examples like when someone uses bitcoin to transfer money, by converting an external currency into bitcoin, sending the bitcoin somewhere then converting it back into an external currency. Whatever loss of value they see from that overall transaction has been transferred into the bitcoin system as a whole.
I am not sure I understand the negative sum game you mention. The first payment miners get is (I don't know the proper name) a bounty, which is equivalent to a gold miner obtaining gold. This can be seen as a built-in inflation, which not only gold but also any fiat currency has. The difference is there is no known limit on this inflation factor for gold or fiat currencies. For gold we assume some gargantuan deposit will not be found and that it will maintain its scarcity. I don't think anyone thinks there is an inflation limit for the US dollar or any other fiat currency. Bitcoin has a known amount limit. (As a disclosure, I own gold, bitcoin, and US dollars.)
The other part of what miners make is a transaction payment. That exists in every other payment system I assume, except I guess physically handing an object to another person, which covers a pretty small fraction of transactions.