I'm just going to keep this simple. Folks can disagree or agree. I will bring this up with my dm. That said, I have been watching lots of videos rating real mideval armors and weapons. These videos are fairly in depth. The Gambison/padded armor is far more agile, than the boiled leather and studded leather. I am not saying that the gambison/padded armor offers better protection.  I will say, that Gambison/padded armor shouldn't have disadvantage to stealth checks. In fact most heavier armors including leather were often worn over gambison/padded armor.

I'd argue Padded shouldn't even exist, it's pretty much a trap option. Hide is too, but at least it's an affordable alternative to Studded early on.

D&D's had a long issue of there generally only being a few armor types that are worth using for PCs and all the others just sort of not.  It's especially pronounced in 5E since the simplified stats mean that there aren't many differences between armors of the same weight class aside from the armor class and whether or not they give disadvantage on stealth checks.

Is gambeson underrated and oddly statted? Yes. But it's also sharing an equipment chapter with, to name the first examples that spring to mind, one-handed longswords and generic all-purpose shields that are somehow still useful even when you're in full plate. It's not that it doesn't bother me; it does, a little. There's just too much to fix one item at a time, especially when game balance and reflecting fantasy tropes an average player will be familiar with from other media are both likely to come before any sense of practical accuracy in a context like D&D.

Light armors are a general mess - padded/gambeson shouldn’t limit stealth, leather should, it isn’t motor cycle leathers it’s cuir bouli and stiff and hard ( leather and felt caps were treated the same  to make the first miner’s hard hats.). Studded leather isn’t real it’s a misunderstanding of a light splint/Brigandine armor that should be a medium armor.

At one point several editions ago, ANY armor had a penalty to stealth.  Thieves (they weren't Rogues yet) all aspired to near nakedness for the sake of stealth!

D&D is not about simulating, it is about representing.

Padded leather armor represents more than just a traditional medieval gambeson. It represents an entire group of armors that include the gambeson, but also includes any kind of armor that is made of quilted layers of cloth and batting. Some of that might be squeaky, or bulky, or somehow avoid being noticed by moving quietly and hiding behind things.

Nothing says that you cannot make it more narrow in your game, but be aware that the rules are not about a specific kind of armor from a specific time and place -- they cover an idea, of a certain type of armor that is padded, but makes stealth difficult. They offer examples that might be familiar, but they are simply that -- examples from Earth's history. Not all worlds share a history with Earth, or have the same functional basis or intent or design.