Why is the growth of GDP so important? I understand that we want living standards to go up in the aggregate, but if the economy produced the same amount of goods/services per year for, say, three years, would that be catastrophic in itself (i.e., aside from policymakers and economists freaking out about it)?
The event that you are referring to is what economists call recessions. During recessions, GDP drops or grows significantly below normal times for a period of usually less than two years. Recessions are dreaded by many but not all economists. Robert Lucas, a Nobel Prize winner, questioned long ago the idea that short term economic fluctuations are really costly to society. He suggested that the public may not be willing to pay much, say in terms of higher taxes, to get rid of economic fluctuations. Thus, you are not alone in thinking that we should not worry too much about short term economic fluctuations.
The dominant view, however, not only among economists, policymakers, politicians, and the public in general, seems to be that short term economic stagnation is bad, and how bad depends on the exact number. A recession of the type you are referring to, of zero economic growth for three years, would be considered severe, probably a depression. To put some perspective, all postwar recessions, excluding the last one, lasted less than two years if one uses the metric you suggest of just looking at stagnated GDP.
Why are recessions so feared if they are typically short-lived and don’t seem particularly costly in terms of aggregate output, or GDP? There are multiple reasons. A key reason is that the unemployment rate significantly jumps during a recession which signifies harsh times for many families. For example, the unemployment rate during the recession of the early 1980s, which satisfies your metric of less than three bad GDP years, jumped from around 6% to almost 11% at the top of the recession. Perhaps more importantly, the unemployment rate typically remains significantly high even after GDP has recovered and it only returns to its pre-recession levels more than 4 or 5 years later, if not more.
Another reason to fear recessions is that they disproportionally hit sectors of that are central to the long term strength of the economy such as investment and research and development sectors. This is because during recessions individuals and firms are quick to postpone long terms projects waiting for better times. A key reason for postponing investment decisions is that recessions are highly uncertain in terms of their duration and severity. Even if the typical recession is short-lived and mild, there is always the chance that the approaching recession is the bad one. Yet another reason to fear recessions is that they affect electoral results. Presidents and other elected officials are less likely to be re-elected in bad times. Thus, GDP recessions may temporary but they seems to have a more lasting effect on many individuals, often the most vulnerable, on firms, often investment and R&D firms, and on governments and institutions.
Mild recession are like tornadoes in the Midwest. In some sense they are not like hurricanes that destroy vast areas, but are more localized dreadful events. Still, the fear of being in the path of a tornado is real and terrifying enough that meteorologists and the public in general are willing to accept some degree of frenzy to save some lives.