I know you've stated that hey, it's not necessary to have a Pokemon evade every single attack. But the problem with that is...well, weavile would be able to evade slower Pokemon like Slowbro, but taking in your previous stat into consideration(endurance), then that would make the likes of Slowbro/Slowking literally untouchable. Think about it, unless Weavile uses Night Slash, it cant make a damn dent in those two Pokemon, and they wouldn't even have to use speed to crush Weavile, they could just use a move like I dunno, Surf to KO it really. I'd say it's pretty damn hard to dodge a tital wave coming for you, but that's just me. XD
But really, i think that the stats that we have now are balanced enough as it is. you'd have sturdy Pokemon being literally untouchable by your standard neutral moves, since they have ridiculous endurance, and then you would have the Pokemon would high evasiveness being untouched by moves that require physical contact, so I do think that it'll be quite a bit unfair, and it'll definitely make competitive battling more of a struggle than it is. For example, you have a Pokemon like....Staraptor, right? Alright, we'll take Staraptor as an example. It's a pretty fast Pokemon, let's give it a good level of evasiveness.
And then let's try it's opponent. Luxray for good measure. Of course, taken into account that Luxray is a slower Pokemon than Staraptor, it's a bit of a problem. Not only this, but if you're applying that logic, then Luxray can't really touch Staraptor much, because Staraptor is a bird, and it can outspeed pretty much just about every single move that Luxray could do it, bar thunder/thunderbolt, and Lux is more of a physical attacker anyway, so it already has a disadvantage from the start.
Ultimately, I feel that I wouldn't be very comfortable with these changes because it would give certian Pokemon a -huge- boost while it would just bump down so many other Pokemon, making them have some sort of unfair disadvantage against those who have higher speed, and those who have higher speed having an unfair disadvantage against those who have extremely high endurance. It'll just really make things a lot more complicated than it should be, which is how things shouldn't be, especially when you consider that kids play these games, and the last thing that we need is for them to be throwing their 3DS' at the wall because (insert 6th gen pokemon here) has higher speed and, by mere luck, keeps dodging the opposing Pokemon's attacks. Even if it's 1 in 4, you have to admit that 25% is still quite a big number in terms of battling, and can make a -huge- difference in the long run.
I think you missed the point I was trying to make, with your example of the Weavile and the Slowking the point I was trying to make is that in the anime they are quite bulky even to moves which are super effective against it so why can this not be reflected in the games. I do enjoy the battling in pokemon games as it is but I just find that I am often bored and use the same pokemon over and over so adding in new stats would help bring some of the lesser used pokemon into action. As for your example about the Staraptor and the Luxray did you never consider if a Luxray was using a move like thunderbolt? Sure the Staraptor is fast but 9 times out 10 a thunderbolt is going to hit a bird pokemon and hit it hard.
What I was trying to achieve this this thread wasn't to get people to start wanting new stats but just to think about the options which Game Freak has to make the game a little more fun and technical by making it more like the Anime which if we're all honest has some of the greatest battle scenes in a TV show let alone in a kids show. I may have miss wrote what I was trying to get across I wasn't speculating that Bulky pokemon like Slowking be untouchable just more bulky when it comes to being hit super effectively some pokemon like Blissey for instance wouldn't get a high endurance because even though it is a big bulky pokemon endurance would work of a pokemons Sp. Defence and Defence but be a completely new stat meaning that just because a Rhyperior is Defensively bulky it doesn't necessarily mean it will have high endurance because it has a low Sp. Defence, pokemon who would have a higher endurance would be pokemon like Mismagius who has a well rounded Sp. Defence and Defenc mixed in with it's low HP means it would likely have a higher endurance than a Rhyperior because it has the ability to tank hits but not the HP to do so.
Man I am loving this intellectual debating over the past few days! :D