Having played a lot of new White Wolf stuff, my group and I hit a little wall. Namely, that we need so many d10 to play almost any more than human character that the system becomes annoying. Players often roll 15 dice and get no successes at all, and then a 4d10 gunshot will miraculously score 10 or more successes and blow one of them away. Now, I understand that a lot has to be said for GM control, but at this point most of the time we are resorting to GM interference to keep things running at all.

Thats the background. Now for my idea for a solution.

Create a simple, storyteller like system. I am looking at the old Iron Gauntlets game by Politically Incorrect Games. I want to strip out the system, using stats for a dice pool, keeping the pool relatively low in number, and skills for the target number required to generate successes. However, I know no one who has used this system before. It seems sound to me, and with a few tweaks, removing some useless and crunchy things, like flairs, threads, and all the other things that seem pretty tacked on, it would work.

For example, Jim has 4 strength, so he rolls 4 d10 for strength based actions. He has the skill Brute Force at what is essentially a rank of 5, so he gets a success for any roll of 5 or lower. Jim is pretty good at bashing through things, and can get a couple successes routinely for doing so each round.

My question is, is there any inherent problems with this (seemingly) unknown system of resolution?

In theory it does not seem bad. What happens when a character has a 0-1 in a stat? They will almost never succeed.

My suggestion would be start with normal storyteller difficulty with no exploding dice. For each rank in skill drop down the amount needed by 1 until it is 5. At the last level or two add in exploding dice.

The system you described is used in a game called The Riddle of Steel by Driftwood Publishing. That game has other basic rolls in addition to the roll you have described and it is referred to as a Skill Check. I have played Riddle of Steel a little bit and these rolls did not seem problematic at all, but on the other hand the game had five different types of rolls to cover different situations.

I personally like simple systems so this kind of a system appeals to me.
I can come up with two questions I think you need to think about in addition to the problem of attribute level 1:

How do environmental factors or equipment affect the roll? Do you add or substract dice based on the situation? This might be a problem with small dice pools, since a +1 or a -1 modifier to a pool of (for example) 1-5 dice makes a big difference.

What about situations where there is no appropriate skill for the roll? If you use a huge number of skills this problem can be dodged, but it can be tiresome. Another way is to use wide-ranged skills (eg. Athletics covers lifting, climbing, swimming, holding your breath etc.) but it might be quite difficult to come up with a small number of skills that cover everything.

A friend of mine did something similar. He made the minimum stat you could have give 2 dice, to avoid the low probabilities of 0-1 die. I don't recall how he figured the target number for success, but I do remember that environmental, situational, and equipment factors added or removed dice.
On the minimalist skill front, I use a modified list from Microlite20, which only has four skills. I use the base skills for all the typical adventuring actions then let players select technical skills.

and then a 4d10 gunshot will miraculously score 10 or more successes and blow one of them away.

such a system can work, the problem is when it seems random

playing shadowrun 2nd edition it was annoying when sometimes I needed only one success but needed to roll a 9, and other times I needed only a 4 but I needed 5 successes.  Example: damage you roll strength dice and each success is a point of damage.  For perception you roll perception dice and you just need 1 success to spot the target.

For that you want lots of dice for strength, but for perception you want higher skill.  It got confusing both creating a character and playing.

Maybe you could have "power" be the attribute and "skill" be the target number.
So for an attack you roll the strength dice, and your combat skill is the target number.  Each success is a point of damage. 
To persuade someone you roll your charisma dice, and your persuade skill is the target number.  Each success is a point of persuasion.