An example of a nice compromise would be the macOS menu bar. Most status icons are monotone, which allows the ones with meaningful color differentiation to shine through without being drowned in the noise or increasing user fatigue.
Citation needed. What is user fatigue? Can it be empirically measured? Fatigued by what? Too much color? Lack of shape? Too much contrast? Lack of contrast?
When is the last time you were "fatigued" by icons?
Without hard facts the expression is just a wishy-washy way to promote a personal taste.
I'd be surprised if squircles reduced user fatigue though - I think a good adjacent example is Googles new icons with the colours that all look similar. Users were complaining immediately that they had to look harder to find the correct app.
Another thing we use everyday - fonts - have differentiated visual spacing and shapes make them easier/quicker to read. So it would make sense for this to apply to icons that serve a similar purpose on a smaller scale.
For a moment there, you almost had us thinking that you were interested in learning something.