STR 9 (-1) DEX 14 (+2) or 15 (+2) CON 10 (+0) INT 11 (+0) WIS 11 (+0) CHA 11 (+0)

Skills Perception +2, Sleight of Hand +4

Those seem like good stats for a fat lockpicker.

+0 dex mod,

expertise in lockpicking (ranges from +4 to +12 by level)

A feature, to mimic Gloves of Thievery +5 to sleight of hand and lockpicking.

They're getting +9 to +17 bonus, while still having a Dex score of 10.

Seems simple enough. Although if this is a PC, may want to consider similar bonuses to other characters for their defining skills.

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As a DM this is totally something I would let my player roleplay out by setting much higher DCs for anything physical that needed a dex roll (dodging etc) and then lowering the dc for things requiring nimble hands. 

 ^^ This right here.
That's what being a DM is all about, not just someone to enforce mechanics.

 How do you set a lower initiative for a character?

Seems to me the whole point of ability scores is so that you *don't* have to set different DCs for every character trying a thing. The DC describes the thing. The ability scores describe your ability. The modified roll combines the two.

 Disadvantage on Initiative checks.
It's not like you are punishing the player... this is EXACTLY what they are asking for.

And you don't have to necessarily set a different DC, just use the binary Advantage/Disadvantage that is already available to you.

 

But why do it this way? Having to apply many special modifiers, forever, for that PC sounds really annoying.

Can you discuss why this would be better than simply overriding the characters sleight of hand? Dex is used for a lot, and almost all those things are things this character would be bad at. This character only would be good at things that involve manipulating small things with their hands. So having an overridden sleight of hand, coupled with making sure to do lock picking checks with sleight of hand, sounds easy and perfect.

I'm not worried about punishing the player. It just sounds a bit harder than the other way.

I would do it this way because that is exactly what Disadvantage is for, and it’s not “many special modifiers”, it’s one simple mechanic that the player can “decide” to use when I call for a DEX ability - it’s all in his hands. As the DM I no longer have to even think about it.
You can do it however you would like, and no way is better than the other, I’m only stating what I would do in this situation based on the type of DM that I am and the kind of game I run... I am not you however, and I’d never tell someone “what you should do”, only “what I would do” (I have a wife for that).

But man, I’m not discounting your idea either. In the past (40 years playing) I’m sure I’ve done something exactly the way you are describing. I’m just giving other options using defined mechanics - you never know when a simple houseruled change like this can slap you in the face, so it’s good to have other ideas.

 Advantage and disadvantage are already overused in DnD it’s why as a DM when I give dm inspiration it is a D6 to add to any roll as opposed to advantage. 

 That's how a universal system works. Saying it's "overused" is like saying "d20's are overused and that's why my players roll 2d10 for attacks and 4d6 drop lowest for saves." I mean you're welcome to do that, but imposing A/D to handle this is exactly how the game intends it to be used.