And, by picking Persuasion as something you're really bad at, you ARE the type of character who would insult the duke because it's what your character would do.
If you're bad at Persuasion, then you'll probably end up insulting the Duke
if you try to persuade him of something. But you also
know that you're bad at persuasion, and that there's a good chance that trying to persuade the Duke will not go well. Since you know that, it may well convince you to just shut up and let the Duke have his way, since you know you're out-gunned in that fight.
I'm boggled by "characters granting themselves inspiration" part. It would be picking skills at character creation, and they need to come up in a meaningful way.
For another example, a lot of nerds are incompetent at climbing, because they're fat and lack upper body strength. But they also
know that they're incompetent at climbing, and as such, they're unlikely to choose to climb anything.
That's what I mean by granting yourself inspiration. You know that climbing is unlikely to go well (because you are aware of your own deficiencies), but you also know that choosing to do this thing will give you inspiration (because that's how the world works). It's not something you can just trigger, whenever you want inspiration - it actually needs to come up on it own - but it's your choice about whether you want to do the thing and thereby grant yourself inspiration.
I suppose you could also find yourself in a situation where you have no alternative, but that's not super common (in my experience). If guards are chasing you and you need to climb a wall to escape, then disadvantage only matters on the die roll if failure would have been a possibility regardless, and I'm not going to blame
myself for rolling poorly in a situation I have no control over.
I mean, if I need to roll a 10 to escape - which is a best-case scenario, really - then failure is a strong possibility
regardless of whether or not I have disadvantage on the check. That's just the nature of d20. The disadvantage mechanic will never allow something to happen that wouldn't have already been a possibility without disadvantage.