Spending money to create currency does not add any value to society. (Or at least, it is a horrible investment, compared to spending 8 cents to print 100 x $100 bills)
And yes, there's more costs to printed currency then just printing bills. However, it does not come anywhere close to the value of the currency. The same cannot be said for BTC. There's 1.2 Trillion printed USD in circulation. I strongly doubt that the cost of printing it is anywhere approaching that amount. There is a billion $USD worth of bitcoin in circulation - and the amount of resources expended on producing it is a much larger fraction of the monetary base.
BTC is not valuable because it is definitive proof of difficult work - its valuable because it is 'difficult' to create, which protects your BTC against inflation/counterfitting. There is nothing inherently valuable about doing difficult, pointless work (Or any work - really), that provides no value to anyone - except shovel manufacturers.