How to compare many bandpass filters (for their signal propogation delay)?
For a linear phase FIR filter, the group delay will be about the same delay as what you have defined to be a rectengular envelope delay. 

For a nonlinear phase IIR or FIR filter (like the Butterworth), the envelope delay is associated with the dominant time constant of its impulse respone $h[n]$. 

Indeed what's happening in your plot is that the filter reaches its steady-state response, after the transients decay. And that would happen after a few time constants. For the relation between group delay of the filter and the time constant, it can be said roughly that, short transition bandwidths in the frequence response (sharp filter or narrow band filter with large time constants) yield more aggressive phase changes which also yields larger group delays at the passband frequencies.

A minimum-phase filter would yield the minimum delay (among the same frequency response magnitudes). A minimum-phase filter's impulse response has the fastest decay and begins more aggresively than the other filters; i.e., it's energy is concentrated at the very beginning of its impulse response samples.