If you are carrying the alternate you try the approach first - subject to reported vis/ RVR being above your company and type minimas. If you don't get in (zero base and either 75 or 100m for Boeing/ Airbus min RVR cat IIIb) then you divert to said alternate.
What Capt Fathom said. They didn't have the fuel to go to ADL. The absence of a CAT3 ILS is not what caused the diversion.
If ADL had had a CAT3 ILS they would still have been required to go to MIA if that was a company alternate and the weather at ADL was below alternate criteria and they didn't have the fuel to divert after an ILS - which they didn't.