As for the maneuverability on raising wages, that's what a minimum wage would be for. No need for companies to let the competition ahead of them because they increase the wages - everybody else would have to do the same.
In the case of your example, if all companies increase wages all at the same time, then you end up pretty much in the same spot you started. Goods and services' prices adjust, companies make less profits, rents increase -- i.e. no actual increase in production (or housing) in the economy that might help the poorest the most. Some industries might be able to pass on the wage bill entirely to customers, others will get squeezed and stop investment and entry of new companies. It all depends on elasticities and the studies that we usually see trying to measure that show that shareholders are not the ones that get hit the most.
If we want the poor to be able to afford more appliances, we need to make those appliances cheaper to produce. And to get there we might need many industries paying people very little, so that we can get off the ground somehow. This example is easy to visualize when looking at China or Bangladesh. But even looking at Europe as a whole it is very clear in my mind that countries like Greece or Portugal still need to cover at lot of ground in this manner. Former communist Eastern Europe countries are already surpassing GE and PT by following broadly that scrip. And even UK and Germany would only be bottom five if they were considered US states (in GDP per capita adjusted by purchasing power). And inside the US there is a massive gap between NY and Alabama. And NY itself is plagued by super high rents because we don't build anymore.
It sounds cruel to imply that a $15/h wage for the lady in the article is not going to help her. It will, but in the short run only. It also sounds impolite to mention that many people in her situation managed to turn their lives around out of that kind of precarity. Many will on account of their smarts, resilience or luck, but many will not. For these the only true hope is that we manage to increase the number of companies and productivity of all companies in the economy. So that we end up with very high productivity industries (read, than produce a lot for low prices) such as oil and gas, car industry, food production, appliances industry, hopefully home-building, “subsidizing” the rest through low cost products and competition of all those low wage, and unskilled jobs. Mandating minimum salaries is like chasing a unicorn.
Side note. In the article she mentions that she is trying to save for a car. Cars in the US are much cheaper than in Europe, and if you go to e.g. Argentina you would be surprised by how much a pile to junk is worth in the second hand car market.