The small attached board could be working as a relay (timer) to take the start capacitor out of the circuit once the compressor is started.
The compressor start time delay could be coming from the unit control board.
T'stat calls for cooling, sends signal to control board, control board starts fan and starts timer for compressor, control board times out sending signal to compressor to start, compressor starts, when compressor motor is up to speed start capacitor board takes start capacitor out of circuit.
Bill and Debbie
2007.5 SILVERADO HD DA CREWCAB LTZ 4X4 ARE CAP
2003 Nash 25S TT
2012 Lance 855S
1996 Coleman Taos pop-up
1970 jeepster,v-6