Why did this extraneous root creep into the solution?

When $x=\frac12$, you have $\arcsin(x)=\frac{\pi}{6}$ and $\arcsin(1-x)=\frac{\pi}{6}$

so $\sin (\arcsin (1-x))=\sin \left( \frac {π}{6} \right)$ and $\sin \left( \frac {π}{2} + 2\arcsin (x)\right) =\sin \left( \frac {5π}{6} \right) $

showing your third line would be a correct equality when $x=\frac12$ since $\sin \left( \frac {π}{6} \right)=\sin \left( \frac {5π}{6} \right) =\frac12$

but your second line would not be a correct equality when $x=\frac12$ since $\frac {π}{6}\not = \frac {5π}{6}$

and it is this use of $\sin$ which creates an equality which was not in the original expression.

$\arcsin (1-x) - 2\arcsin (x) = \frac{π}{2} \Rightarrow x=0 \text{ or } x=\frac12$ is a correct statement, but checking shows $x=0 \Rightarrow \arcsin (1-x) - 2\arcsin (x) = \frac{π}{2}$ while $x=\frac12 \not \Rightarrow \arcsin (1-x) - 2\arcsin (x) = \frac{π}{2}$

$$-2\arcsin(x)=\dfrac\pi2-\arcsin(1-x)=\arccos(1-x)$$

Now using Principal values of $\arccos$,

$0\le-2\arcsin(x)\le\pi\iff0\ge\arcsin(x)\ge-\dfrac\pi2\implies-1\le x\le0$

Now let $\arcsin x=y\implies x=\sin y$

and $\arcsin(1-\sin y)=\dfrac\pi2+2y$

$$1-\sin y=\sin\left(\dfrac\pi2+2y\right)=\cos2y=1-2\sin^2y$$

$$\implies \sin y=0,\dfrac12$$ but $\sin y=x\le0$

The sine function is not one-to-one. When you applied that function you also got

$\sin(\arcsin(1-x))=\sin(\frac{\pi}{2}-2\arcsin(x))$,

as this RHS equals the one you intended. Note the sign change on the right side.

Then $x=\frac{1}{2}$ satisfies

$\arcsin(1-x)+2\arcsin(x)=\frac{\pi}{2}$.

None of your implication steps are wrong

Let's review the concept of implication. In mathematics, an implication is represented by the symbol $\implies$ and when we write $A \implies B$, it means precisely that if A is true, then B is true.

Here is a quote of your second implication:

$$\arcsin(1-x) = \frac{\pi}{2} + 2\arcsin (x) \implies \sin (\arcsin (1-x)) = \sin \left( \frac {\pi}{2} + 2\arcsin (x)\right)$$

This is perfectly correct, since the form of that statement is $A = B \implies \sin(A) = \sin(B)$, and clearly if A = B, they are the same thing, and their sine will also be equal.

Well then where is my mistake? Surely there is a mistake somewhere

Let's summarize what you've proved:

$$\arcsin (1-x) - 2\arcsin (x) = \frac{\pi}{2} \implies x = 0 \text{ or } x = \dfrac{1}{2}$$

This is correct! (bear with me, I'll get there).

Assume my car is blue. Then I tell you the following sentence: "My car is blue or my car is red." That sentence is correct!

See what's happening here?

When we solve equations, we should be using "equivalence" steps, not "implication" steps, because we want to find all the values of $x$ that satisfy the equation, no more no less.

I find it enlightening to look at it this way because we don't lose the benefit of formal math which IMO makes much easier to find mistakes. By understanding well how implications work, it is clear that there is no mistake in the sequence of implications you created.

Of course, the other answers are correct as well, but I'd like to provide this formal insight because I find it very useful.

If you replace the implication symbols ($\implies$) in your attempt by equivalence symbols ($\iff$), you'll have a mistake on your second step, for the reasons other people already mentioned.