It’s not that it “stirs up solids,” it’s that a continuous leak causes the pump to cycle on more frequently. Your pump might cycle a few times a day normally; a running toilet could cause that to be every 30 min, and at that rate you could need a new pump in a year or two instead of 10.

Trickling your faucets (just a drip-drip-drip) for a few days is a small amount of extra wear on your pump in exchange for not rupturing pipes.