A projective map is a function which preserves cross-ratio. So consider a circle $\Gamma$ and a point $P$ not necessarily on $\Gamma$. Will the function which maps $\Gamma$ to the pencil of lines going through $P$ via $X \mapsto PX$ be projective?
I think it should. Consider any 4 points $A, B, C, D$. Let $\gamma_{ABCD}$ be the conic which passes through $A, B, C, D, P$. Then the cross ratio will be preserved, since the map from a conic to a pencil of lines going through a point $P$ on the conic via $X \mapsto PX$ is projective.
Is this true?
For a conic $\omega$ let $(A,B;C,D)_\omega$ be the cross ratio of $A,B,C,D$ with respect to $\omega$, defined as $(EA,EB;EC,ED)$ for any point $E$ on $\omega$ (it is the same for all $E$ on $\omega$, see math.mit.edu/~notzeb/cross.pdf, Definition 5).
Let $\gamma$ be the conic which passes through $A,B,C,D,P$. Unless $P$ is on $\Gamma$ (in which case $\gamma=\Gamma$), $(A,B;C,D)_\gamma \ne (A,B;C,D)_\Gamma$. But this means that in general the function $X \mapsto XP$ does not preserve cross ratio, and is not projective, because $(AP,BP;CP,DP)=(A,B;C,D)_\gamma$.
By way of context, we can define a conic as the locus of points $E$ such that $(EA,EB;EC,ED)=k$, where $A,B,C,D$ are four points in the plane such that no three of them are collinear and $k$ is a nonzero real. When the four points are fixed and $k$ varies, we obtain the whole family of nondegenerate conics passing through the four points (I'm paraphrasing Berger, bottom of page 201). Conics $\Gamma$ and $\gamma$, as defined earlier, correspond to different values of $k$, unless $P$ is on $\Gamma$.
The OP asks whether their argument that the function is projective is correct. It is true that the map from a conic $\omega$ to a pencil of lines going through a point $P$ on $\omega$ via $X \mapsto PX$ is projective. But this assumes that the cross ratio being used is $(\dots)_{\omega}$. But I think the original assertion is asking whether the function is preserving the cross ratio $(\dots)_{\Omega}$ with respect to the circle.