It is a challenge as it is the typical tool, but there are other tools the central banks can use - the asset purchases are an example. But it is the case that in the economic policy, the longer term economic rebalance it is preferred that the fiscal side, that is the direct government spending, of preference in investment in the assets or the human capitals, attack. The pro cycle austerity programs of the past decade since the crisis have not proven to be effective policy - the policy idea and framework that has worked when a crisis was essentially a single country crisis broke down when it became the global crisis effecting all the big economies.
A lesson that there can not be one rigid reaction to a problem in the economic policy, but there must be thinking about the scale.
It is also a lesson that probably the obsession in the big developed country central banks with having the 2 percent inflation was an inflation objective that was one that was too low.
There was too much obsession with a false idea of a hyper inflation risk (warned about since 2008 and never arriving - not there is no such risk, it was not the real risk for the developed economies, the deadly deflation was and even now almost one decade later still is).
The central banks do not closely track or care much about the stock markets, which are too narrow to reflect the overall real economy of a country. That is why all the central banks that are well developed engage in the business surveys and developed the business statistics trying to track the real economy, not the transparent financialized economy. It is a major source of data, the Federal Reserve research in fact, if I want to look at the US economy. Not your stock markets.
They are very mediatised in the Anglosaxon world - and most greatly in the American medias - the stock market indices, but they are not the good economic indicators for the macro economic policy planning. They represent too narrow and too sector biased segments of the overall economy.
The entire point of the low interest rate environments has been that there is on the macro level too much of the savings, not enough of the investment and the consumption.
The central banks have a limited approach so the addressing of the stagnation on investment and the economic growth, which is a complex problem, only via the central bank will always have side effects not desired.
One of those is the hitting of the small saver on the savings account - although in most places the small saver bank accounts are not very important overall. It is the collateral damage.
I think it is just that most people globally have a very poor understanding of the workings of economies and they tend to grab a hold of some easy to follow symbolic number - the stock market is very popular for the Americans, like the Dow, although the Dow has so little to do with the actual macro economic health.
then since they do not understand better, they think that people in the Central Bank are following this same metric, and as it does not make sense to them, they invent stories to make it make sense…
In other places the story is different, but the mechanism of the story is similar.