Not sure if it is the starter problem. Back many years ago, a mechanic told me to try a trick to start my car in the dead of winter. He advised me to boil some water (not for the car) first. Get in the car, and click the ignition key once without starting the car. Get back into my house and drink a hot cup of coffee, say for about 5 to 10 minutes. Then when ready, get back to the car and start the car. It worked every time.

The reason behind that was that the first click would "wake up" a cold sleeping battery. After a short period of time, the battery was sufficiently ready, the second click would start the car.

So, in your case, maybe that is in tune with my former mechanic's theory.