Considering that percentage is higher than margins in quite a few businesses I'd say it has a limiting effect on the economy.
What is the cost of water in the GDP? Much less than the cost of the energy. What would be the GDP without water? Nil. There would be no GDP because we would all be dead.
Also, you somehow equated the complete lack of a resource with a price increase. What would the effect on the GDP be without any energy? Well, we might not all be dead, but a significant portion of us might for various reasons including wars and massive world unrest because of instability. Even if you ignore all that, the economy as we know it would grind to a halt.
For physical goods, most of it. Raw materials in particular aren't expensive except in terms of the energy required to extract them and process them (modulo supply and demand).
For labor it's more complex, but in general falling energy costs drive capital investment in automation (displacing and lowering demand for less-skilled labor) to lower price and capture greater market share (in order to get a return on the investment of capital), lowering labor's share of the final price, and the Jevons paradox drives increased consumption of the lower priced goods (as well as the lower priced energy). Software has been driving the same sort of dynamic for a while, though the way it affects industries seems to be more contingent, 'lumpier', and less predictable (eg. in hindsight, the disappearance of the travel agent as a result of online booking was somewhat obvious, but AirBnB effectively adding a lot more inventory to the market wasn't).
Not sure what happens when software and energy both more directly affect each others' supply and consumption, but the effect on the rest of the economy is going to be... interesting. Smarter energy grids that shift the economics of energy plants and sources at different scales; vehicles, homes, factories, & data centers that can all adjust their energy use/storage to take advantage of spot pricing, energy costs driving the ROI of training machine learning models etc. and thus changing the return on investment in compute capacity (and in software efficiency), and so on. There are a lot of feedback loops, and as "software eats the world", more (unanticipated) feedback loops will be created.
Would a lower energy cost bring wages up? No reason to believe that would be the case.
Would it lower the cost of goods so that at least the working poor could have a better standard of living?
Perhaps, but how much of the cost of a good is the energy, how much labor and raw materials? Would we see even more job loss due to lower cost of energy as it becomes even cheaper still to automate?
(Edit: to be clear, OP wasn't saying lowering energy costs would do anything more than lower the cost of goods. But that statement alone, coupled with the article showing declining wages, etc. suggests OP was implying lower energy costs could reverse or stop the trends shown.)