The Argument for Padded Armor
I've been watching some YouTubers talk about their predictions for future D&D updates recently, and one prediction was the removal of padded armor since it seemed unnecessary to the individual. Sadly I made no note of who this was since I'd love to give an attribution, but I will admit to having two rapid reactions. The first was a little internal chuckle, "of course it makes sense," and then, "oh right, not everyone who plays D&D is all nerdy about armor."
For me D&D intersected with an interest in knights, spawned an interest in Samurai, led tangentially to live action role-playing, and led me into a ten-year hobby in which I read deeply on European and Japanese arms and armor. All of that is about twenty years in my rear-view mirror now, so for a more updated perspective you may want to look at the Shadversity YouTube channel. But here's my two cents.
As an armor that you'd buy to wear all on its own, padded armor doesn't make much sense. But there is a garment called a gambeson or padded gambeson that knights wore under their armor. The garment serves as an added layer of protection against being bashed and as a garment to attach armor to the wearer.
Chain maille, or maille as it was called, is pretty great at keeping pointy things from intersecting with your vital organs, but it is, as you can imagine, quite flexible. Were you to wear a thin shirt underneath it, you may be saved from stabs and slashes, but the powerful blows could still break and crack your favorite bones. And then of course there are arrows. The gambeson gave the knight a layer of protection against the force of blows and some space to slow the velocity of arrows before they could pierce flesh. I've heard stories of uncertain validity about knights walking around like porcupines with arrows hanging harmlessly from their maille.
As technology progressed and plates began to be added to maille and gradually replace it, the gambeson continued to serve its original purpose and became a garment that plates could be attached to. Aside from plate armor's strength, there is a strategy as to how the plates are layered on the knight. The goal is to maximize flexibility and, not surprisingly, minimize the penetration of weapons.
So what is padded armor in D&D terms? I've always taken it to be the gambeson. At some point in the day your players need to remove their armor and get some rest. When they are inevitably ambushed, and I do hope they are occasionally ambushed when their armor is off, your warrior types can take some small comfort in the gambeson giving them a minimal form of protection.