I'm sure I could heal ICC quite competently without any add-ons at all, but I think it would be silly to do so. I certainly don't think I could lead a raid as effectively or alert others to important information about the fight.

The metaphor of a crutch is extremely inappropriate. People walking with a crutch usually walk more slowly than people who don't need crutches. This is not at all the case for healing with addons. If you feel addons are somehow unfair then compare them to performance enhancing drugs if you must, but the people in the best guilds in the world use addons. People who heal with addons heal better than people who refuse to use them.

But furthermore, addons don't just makes things easier, they can also challenge you by giving you more information than you could possibly get with the default UI. On heroic Sindragosa I decide where to stand and who to heal and with what spell based on the number of stacks of Mystic Buffet that everyone in the raid has. On heroic Lich King I need to be looking at my environment to determine the relative location of the necrotic plague target and the ghoul pack, as well as shadow trap locations, and without having all the other information I use (tank health, incoming heals, who has shield and weakened soul, time to next infest, whether anyone has aggro on a ghoul, boss health percentage) in one compact box I would just have to make decisions based on less of that information because doing the wrong thing quickly is better than doing nothing at all while you think about it.

Use addons to get more information about what is going on and use that information to make better decisions. If you aren't using addons you are choosing to get less information than you could have. That's not to say you can't be very good, because you can. But most likely you won't be as good as you could be.