As your question astutely identifies, the need for accuracy depends on the rider's use case. In addition to the two you've already pointed out, there's a third common use case: when you get an additional, or replace an existing, power meter. In that case, you typically want to compare across two power meters either concurrently or over time. For example, power meters can fail over time; or, sadly in my case, my bike and power meter were stolen. It's easiest to compare power data over time or over different units if both are accurate.
That said, some use cases, even if you're not adding or replacing a power meter, are less demanding of accuracy than others. Training FTP (functional threshold power), for example, is arguably one of the least demanding uses for power data. Racers trained their fitness and performance for a century before the introduction of affordable on-bike power meters by riding known courses or known hills and keeping track of elapsed time. Starting in the 1970's, riders used heart rate monitors. For this purpose, one doesn't need much accuracy at all--all one really needs is a power meter that is "ordinally" consistent (that is, that maintains consistency in the order of the level of power). A deeper requirement of accuracy is what is called "interval" consistency, so that the difference between 100 watts and 110 watts is the same interval as between 200 and 210, or 500 and 510, or 1000 and 1010. Certain kinds of training for specific purposes (for example, for sprint training, or for determining training load) can require interval consistency.
In general, we've long known how to go faster on a bicycle: more power, less drag, or better tactics. Properly used, a high quality accurate power meter can help you achieve all three. An ordinally consistent power meter can help you with the first, but not always the other two, and the finish line and timing clock don't care which of those three you used.
Besides online racing (your use case #2) and continuity of data fidelity, two other use cases that demand power accuracy are drag estimation, and training load estimation. In both cases, you use the interval relationship between different powers to calculate the quantity of interest. You may be familiar with the common training metric "training stress" or similar metrics. The most common (but not the only) training stress metric is TSS, which is based on "normalized power" (NP). NP is a way to take variable power data from different rides and "normalize" them so they can be compared to a putative "steady state" power. NP uses the L4 vector norm (hence the name) in which the wattage is raised to the 4th power, its mean is taken, and then shrunk by the 4th root. This algorithm depends on interval fidelity.
For drag estimation, we are interested in both aerodynamic and rolling drag. Rolling resistance drag varies linearly with speed while aerodynamic drag varies with the cube of speed, so to be able to properly allocate the power to these (plus the power dissipated in acceleration and climbing) you need accuracy in power measurement.
All of the above being said, at least at the moment, it appears that most riders use their power meters only for training FTP. A handful of riders race in on-line leagues (for example, Zwift; but there are others) and they need to use approved power measuring devices and dual record their data to ensure accuracy and fidelity. Fewer still do drag estimation, though it is quite common among pro riders and especially among those who attempt world records or Olympic medals. Since the dominant use case is training FTP, and training FTP is one of the least demanding use cases, most riders can get by without much need for data fidelity. This is a good thing, and has democratized access to power measurement tools.
