When a parent that is EV trained is used for breeding, the IVs it passes are based partially on its EV training as well as its IVs. Essentially, it would work like this.
When a parent passes down a particular IV, it looks at its EVs in the same stat as that IV. That IV is increased or decreased by a certain amount from what the parent would pass depending on the EVs, ranging from up to a -31 penalty at 4 EVs to up to a +31 bonus at 252 EVs. Basically, every 4 EVs in a stat is one bonus point when passing that IV, starting at -31. (0 EVs results in no IV change, to keep it reasonable for chain breeding.)
For example, suppose you breed a Fletchinder with 100 ATK EVs and 28 ATK IVs. If it passes its ATK IV, instead of passing 28, it will pass 21, since 100 EVs in a stat results in a -7 penalty on the IV. Let's say you go grind up its ATK to about 180. Now, because of that boost, it gets a +13 bonus to the IV. Of course, the IV cap doesn't change, so it still would only pass 31.
The practical applications being that you could more effectively manipulate IVs if you want a specific non-31 one. For example, if you're going for a particular Hidden Power type, it would become considerably easier to do so.
Why are all of these so awful? Breeding is half-decent now.
Your example, my pokemon:
252SPA 252SPEED Charizard in daycare. He's only going to pass on x/x/x/31/x/31 IVs, in spite of the fact that he has 31/x/31/31/31/31 IVs.
...except that's not how the suggested change works. It can still very well pass 31/x/31/31/31/31. It depends entirely on what it passes and what the other parent passes. In this case, the only thing that can actually be different is if it happens to pass the stat that has your extra EVs, in which case that one will take a huge penalty.
The idea is that it would allow you to specifically train stats exactly to your specifications to make it easier to pass specific non-31 IVs.
Let me get this straight. If you have a 31/31/24/30/31/2 parent, tailoring the EVs on it purposely could potentially result in a 31/0/31/31/31/31 baby? Something like that?
That sounds ok.
Existing breeding mechanics wouldn't be lost or anything. This would just be another way to influence IVs. If anything, it makes breeding even better. You can still easily get your perfect 31s, but then you can also more easily get the Hidden Power types you want.