Let get ALL Fact Right
ERC = Electric Rule Check .... Check is all pin connected, and is the pin topology going to the right another pin topology (input to output, and not output to output) and many more, see it when you create a component, or edit some component in pin directions.
DRC = Design Rule Check .... it only looks for is all pin get a trace and is the isolation between track/via/danger zone/outline/pcb high track crossover... and so on. to make sure the program can make a Gerber file
Spice = a program that calculate electronic signal, not how it shot work but WYSIWYG, this is done with only the Math where all model in you .sch require spice model Info in the model, where you as designer shoot type all relevant Parameter for this model in the specific model
Kicad Was made to make opensource or private project PCB and northing else, so do not expect some part with Spice