How often does he try, who knows?
Is he in a party with one or more professional sneaks who could give him pointers (reasonably likely), who knows?
Exactly. And
if he's paying attention and trying to learn how to sneak effectively, in mechanics he should (somehow) gain proficiency in Stealth. More to such a point, things like using downtime activities to "gain training" in a skill should be part of the game--but all they give us is learning a language or tool (in general).
The point is it doesn't matter how much effort he puts into it in-game. It doesn't matter who his party members are or what he does with them. Without explicit investment of an ASI or character level(s), they stay just incapable at sneaking past a goblin scout from level 1 to 20.
Again, exactly.
For example, I've played golf on and off for over four decades. Frankly, I'm really not much better at it now than I was back in my teens. I'm probably better in some ways, worse in others. My point is I could never claim I've practiced or trained in golf in any consistant fashion--- I haven't invested into it. So, I don't expect to really be any better.
A paladin in heavy armor should rarely be trying to stealth (without serious help!). He should be like me with golf. Maybe slightly better, but probably not even to the point of getting a +1 bonus.
This is why, for example, if a
group of PCs is trying to stealth, I prefer the group check mechanics. If half of them succeed, the party does. This is how I assume the skilled PCs are giving pointers and helping the unskilled.
I'm kind of ok with him remaining bad at stealth, but I see an argument for some kind of growth to reflect that hey, maybe they didn't train, but they at least picked up a couple tricks along the way.
I'd expect the same thing with every skill. Your barbarian may not be a lore master, but he has been in a disproportionate number of ancient temples compared to the broader population. Your wizard may not be ready to take the mat vs Royce Gracie, but they've had to wrestle to get away from creatures plenty of times. Your ranger is no diplomat, but they've often been a party to negotiations with royalty, bandits and dragons.
I don't know what the right size for such a bonus would be, maybe, just enough to keep the target die roll somewhere around consistent with scaling DCs.. but "more than nothing" seems like a reasonable start.
Oh, I see the argument for it as well. IMO and IME, usually the bonus is so minor it really has little impact (so why bother?) or is big enough that it makes too much of a difference, and makes high level PCs without proficiency just as good (or better) than a low-level PC with proficiency. For me, anyway, either is not an acceptable option.
Granted, I don't care if someone else does implement some sort of improvement system, it just isn't my thing in general.