Aegislash is bonkers, especially if you have 0 speed IV and a -speed nature. You can use alternate between powerful attack like Gyro Ball and using King's Shield, which changes you back into shield form with priority. Generally, you'll never be taking an attack while in blade form. Thus, you effectively have base 150 defenses and attack/spatk, which is absolutely nuts.
E: Doublade has only 110 attack and 45 spatk vs Blade Aegislash's 150. Eviolite Doublade does, however, have higher defense than a similarly invested Aegislash, but even a fully invested SpD eviolite Doublade will be about the same as a 0 SpD IV, 0 EV, neutral nature Aegislash. And then the Aegislash gets to hold an item, which is huge.
On paper, Aegislash is much much stronger than Doublade. But in practice it isn't so simple.
Gimmick Pokemon like Aegislash are very easy to deal with for experienced players. The problem with Aegislash is that it trades its stats and while you think with perfect play he will always have his best foot forward, reality isn't so simple.
See, King's Shield isn't an invincible wall. Unlike protect, KS doesn't protect against status or other such things (burn and paralyze completely screw over Aegislash). Additionally, it gives a free move to your opponent. Not to mention if you decide to try and mindgame your opponent by not shielding, you risk committing suicide with your now really low defense.
This doesn't even include things like multi battles where Aegislash has to contend against multiple speeds.
-----
On the other hand, Doublade is just consistent. He's basically just a wall that's really hard to kill and will do a lot of damage before dying.
It doesn't matter whether you attack as your stats are still the same.
I personally prefer that simplicity to Aegislash. It sounds like it's easy to deal with because it's so simple, but in my thousands of battles, I'm more annoyed by Doublade than by Aegislash. Generally because if I see a Doublade, it means I'm fighting an experienced player.
---
But yeah, none of this matters for Solo play unless you're playing one of the difficulty mods.
You can beat the game with a level 20 metapod if you wanted.