Current sensing power supply for hobby servos
An alternative approach might be an IC like MIC2545A, Programmable Current Limit High-Side Switch. To address your question points:


The device replaces the current sense resistor, goes on the high side, and also provides a current sense flag output, current limiting, and a logic level MOSFET switch with a 50 milliohm RdsOn, in an 8-pin package (6 actual pins).
An optional series RC, parallel to the current limit set resistor, provides the smoothing out / low pass you need - see datasheet.
The steady-state current limit is set by a single current limit set resistor, to a range of 500mA to 3 amperes, although with only a +/-20% accuracy in your desired range. Also, as the IC is a current limit switch, your requirement for limiting is met.
Not required, the device does this internally.
Yes: This may not be the optimal part for your specific need, there may be similar, cheaper devices that meet your specification. Experts in this community would suggest better alternatives, I trust - an n-way array version might exist, so you can address your 8 motor requirement with 8/n of the ICs.


As both through-hole DIP and 8-pin SOIC packages exist, you have both prototyping and final product options.