I don't think the grapple half-speed movement rule applies.
In a grappling sitch, the two participants are actively opposed. An unconscious person cannot oppose the effort to drag them. And they're not so heavy as to reduce your speed so severely I feel.
So me, it would take the "interact w/object" action to grab the limb of an unconscious person, then normal movement, leaving you with an action and bonus action (perhaps a dash to get to cover).
Now, game wise, at my table you tell me you want to use your turn to drag your ally to safety, and that's all you're doing - you get it. It happens. Action economy/rules take second seat here, for me.
Why? 1.) because you've declared an action and intent that ought to be reasonably possible and don't require strict adherence to rules to carry out, and also don't require rules to resolve any ambiguities.
2.) because this is the heroic sort of thing I want to see more of! And the rules shouldn't stand in the way of that sort of play or inhibit that sort of heroism with unneeded complexity.
-Brad