Auto start/stop has been around for long enough that there is very solid data on whether it causes premature wear and failure of starters and other components.
Short answer: No, it does not. It's an internet myth.
By one estimate (from
Toyota, who are not exactly known for unreliable cars), you would need to replace the starter after 384,000 cycles. That's more than 21 starts/day for 50 years.
The purpose of the system is to save fuel, that is to say, your money. Again, there is solid data on this. Using it saves you 7.27-26.4% fuel (
source).
Over the lifespan of the car, that's going to be thousands of dollars. Way more than what a new starter would cost, even
if the system did destroy it prematurely, which it does not. Sounds like quite a lot of money to be leaving on the table to me.