I have over 12,000 hours in Cherokees in this area and never had carb icing, in the Archer the air intake is under the cowling, through the air filter then alongside the engine to the carb so is pre heated, unlike some aircraft where it goes straight through the air filter at the front into the carb therefore much colder. POH says to apply it only if there is an indication of icing. Use of carb heat is not one size fits all, and even different Cherokees have different intakes and filter positions, but all preheat the air intake in some way.
I'd say carb ice was extremely unlikely, but the emergency check list says to apply it if you have an engine issue so that, as you rightly said, there is still some heat around the engine. Another kind of issue preventing fuel from getting to the carb is possible. In these aircraft if your throttle cable lets go a spring will take it to full throttle, so it wouldn't be that.