D&D is a turn-based game; however, it is the job of the DM and Players to create an immersive reality. That being said, in reality, if someone disarms you of your blade, do you really think you could just "use a free action" and pick it back up, then proceed to attack them, without any consequence?
Think of it like Inigo Montoya. He disarms you, and holds his blade to your throat forcing you to surrender. If you move to take your blade, you'd be taking an opportunity attack with advantage, and possibly, a straight up death blow. Because he had a weapon held to you.
Now, sure, you could meta game and argue that, well, it would require a feat or a bonus action to "hold a blade or hold action or hold to charge" but in reality, it doesn't. It doesn't at all. You disarm them for a reason, and that reason is not to simply let them pick it back up.
At the very least, you're prone to an advantaged attack of opportunity using their reaction against you. At most, they'd be killing you out right, depending on surrounding circumstances (other enemies or allies in the area).
Just something to consider for other DMs who aren't sure what the purpose of Disarming is since it has no useful ruling, think of it like real life. If you're disarmed, you're not just picking it back up and continuing on like nothing happened. 6 seconds is what a turn is, but these should not be "turn-based fights"; these actions are to be narrated as a story, not like Final Fantasy X.