Got too long for comments.
The -3dB point is traditionally regarded as the end of the useful passband, rather than the beginning of the useful stopband. The latter is too dependent on specific needs to have a single universally applicable definition.
One aspect that makes it most useful is reciprocity : because the R and C components (or L and R) of impedance are in quadrature, making their magnitudes equal gives you sqrt(2) voltage loss, 0.5 power. Any other definition would not have this property. For example, interchanging R and C gives you a high pass filter with the same nominal cutoff frequency. Any other definition of cutoff frequency would give you a different frequency for the transposed filter! Thus -3dB is a uniquely useful definition.
It's a single reference point. Given the 3dB point and a little more info (filter order, type e.g. 4th order Butterworth) you can tell where other characteristic points : 1dB flatness, 60dB stopband etc are.
Or you can work backwards : if you need 40dB attenuation at 1 kHz from a 2nd order Butterworth LPF for example, you know from filter design sources that a 2nd order filter has an ultimate slope of 40 dB/decade, and in the special case of a Butterworth filter, intercepts the 0dB line at the 3dB point, so the 3dB point will be 1 decade from the start of the stopband, i.e. 1000Hz / 10, or 100Hz. If you need 60dB attenuation, the -3dB point becomes 1.5 decades below 1000Hz, or about 30Hz. If that's too low a frequency, you need a steeper filter, such as a higher order one.
Filter design by scaling and similarity, with reference to the -3dB point, has a long history, and accumulation of experience and literature behind it.