Car: 2011 Mazda 3 hatchback (rebuilt title)

Issue: Car has no power steering when temperatures are under 23F/ -5C ish. It works perfectly fine when it's above that temperature. At the borderline temperature, sometimes when the car has been running a while and is re-started, the power steering works again. Under -7 C and there is no chance of it working. The rest of the car drives just fine other than the power steering going out. Mechanic hooked their machine thingy to it and it shows as a Total Steering Pump Failure but like I said, car works fine as soon as the temperature goes back up. Has been going on for 3 years undiagnosed as the temperature rarely dips that low, but this year has been cold and I don't feel safe driving it anymore.

Tried: Assumed the fluid was freezing for some reason. Performed a flush and fluid replacement but the issue is ongoing. Battery tested good this summer. Mechanic advised to change the entire pump and lines for $1200-1600 but cannot guarantee this will solve the issue.

Has anyone ever heard of this before? I think it's an electrical power steering pump - wondering if the cold could zap the battery so badly that it can't power the steering pump? I don't want to spend $1200 on a new pump unless it will actually solve the issue.