I'm opposed to these abilities being allowed mostly for the reasons
jas61292 mentioned, but I am not as staunchly opposed as I have been in the past. I am not opposed to Huge Power, Pure Power or Fur Coat for any reasons of them being overpowered, and I don't think I ever was opposed for those reasons. I am opposed because these abilities are a problem for our process.
It is almost impossible to discuss and vote for these abilities during the primary ability step (and yes, these are definitely only primary abilities) without poll-jumping all to hell about stats, OR without having a ton of very unhealthy unspoken assumptions that everyone is carrying around in those discussions, but no one can voice them because poll-jumping isn't allowed in CAP, for good reasons.
Yes, the same general thing can be said for many other abilities, in that they have a marked effect on later steps, and we can't discuss those later steps during the Ability step. But Huge/Pure Power and Fur Coat are so completely directly tied to specific stats -- they are almost unique in the game in how they convolute the normal CAP process.
For example, when we discuss and/or select Intimidate for a CAP pokemon, we make a significant bump in that pokemon's ability to withstand Physical attacks. There are lots of other mechanics associated with the benefits and drawbacks of Intimidate too. But when Intimidate is being discussed, every intelligent discussion participant knows we are probably going to account for the ability choice later when we discuss the pokemon's Defense stat later in the process. But the impact of Intimidate is fairly indirect in its effect on the Defense stat. The ability only impacts effective Defense against the opponent on the field and is removed versus switchins, the positive impact to Defense doesn't apply to moves like Psyshock and Secret Sword, the ability could actually SUBTRACT from effective Defense if facing a Defiant opponent, etc etc. The existence of all these caveats and special situations with Intimidate don't stop most of us from still generally considering Intimidate to be a physical defensive buff, but no one considers Intimidate to be almost exclusively a buff to the Defense stat of the pokemon in question. There are so many other impacts from Intimidate to the pokemon and to the entire team that has an Intimidate team member -- that the impact to effective physical defense for the CAP pokemon itself, is simply one of many things to be considered when discussing and choosing Intimidate as an ability.
Huge Power, Pure Power and Fur Coat are a completely different story. The impact of these abilities is direct, focused, and effectively immutable. There is no doubt, ambiguity or subtlety in determining the effect of these abilities on the CAP creation. They double the Attack or Defense stat of the pokemon with the ability. Period. Not the "effective physical attack" or "effective physical defense", like in the Intimidate example earlier. Mechanically, the game takes the pokemon's Attack or Defense stat and multiplies it times two. That's it. Yes, there are moves that can remove abilities, and things like Mold Breaker suppress Fur Coat. So I am not saying Huge/Pure Power and Fur Coat are mechanically exactly the same as doubling the pokemon's actual Attack or Defense stat. But the impact of these abilities is incredibly specific to stat values only, and their impact is so rarely negated for the pokemon with the abilities, that there is virtually no ambiguity or question as to what effect will be achieved by giving the abilities to a CAP pokemon.
A CAP poll that selects Huge Power, Pure Power, or Fur Coat is basically a direct vote for a high Attack or Defense stat of the pokemon we are creating. It is not a vote to "make a potent physical attacker" or "make a good physically defensive pokemon". It is a vote to give the CAP pokemon a high Attack stat number or high Defense stat number. I say "high", because there is no reasonable base stat number that is low enough for the pokemon to NOT have a high stat after it is doubled, taking EVs and nature into account. Even craptastic base stat numbers like 60 base Attack for Medicham, with Pure Power it had one of the biggest Attack stats in the game for several generations.
Again, I will reiterate that I DON'T think the effect of these abilities is overpowered inherently, and that is even more pronounced in recent generations where sky-high stats are pretty much table stakes for inclusion in the OU tier. In fact, part of the reason I have softened my opposition to these abilities is precisely because of stat inflation in the game. CAP has to optimize the hell out of stats these days to make our pokemon compete with the vast number of Legendaries and Mega Evolutions that fill the ranks of OU and even lower tiers. CAP still tends to frown on stat spreads with more than 580 BST, so we have to work hard to make our CAP pokemon stat spreads effectively competitive. A pokemon with Huge power or Fur Coat certainly gets a LOT more stat room to work with under that soft 580 BST limit.
But again, this still gets back to my point about the reason for these abilities -- they are nothing but a portion of the Stats step masquerading as the Ability step in the CAP process. If we need more BST room to make effective stats for our pokemon, let's just increase our BSR/BST expectations. If we want our CAP pokemon to have a big Attack stat or big Defense stat, then lets just do that -- but discuss it openly in the Stats Discussion and vote for it directly in the Stats Poll. Don't allow it in Ability stages where the direct impact to the stats is exact and undeniable, but direct discussion or acknowledgement of stats is DISALLOWED by rule.