Hi. You certainly can use the radiator's temp sensor's switch. But remember, it will work when the coolant gets hot, but not always when your A/C needs the extra airflow, which (you wrote) was the reason for your auxiliary fan installation).
A better way to ensure that the fan always works with the A/C is to tap the wire to the compressor clutch. Attach that to a relay (so the clutch's wire isn't burdened by both the clutch and the relay), and then power the fan through the relay. This way, the fan will go on whenever the A/C cycles on.