How could one argue that centripetal acceleration points inwards?

If we consider centripetal acceleration #a_c = v_T^2/r# from the equation:

#F_c = ma_c = (mv_T^2)/r#

We know that the linear velocity of an object in uniform circular motion (UCM) is tangential to the circle, and when the object is let go, it'll travel in that tangential path. Here is a diagram I drew:

If we draw a tiny right triangle where a small #v_T# is the short side and we zoom into that short side, treating it like a limit derivative, then the radius is approximately the hypotenuse of this triangle. If we let the angle of rotation be #theta#:

#costheta = a_c/r => r = a_c/costheta#

#sintheta = v_T/r => rsintheta = v_T#

#=> a_ctantheta = v_T#

#=> tantheta = v_T/a_c#

With this relationship, we have an opposite-to-adjacent relationship between #v_T# and #a_c#, which means they must be perpendicular at all times.

Since both #v_T# and #a_c# must start from the point of examination (their intersection), a vector points away from such a point, and the UCM involves a rotation axis crossing the center of the circle, #a_c# must point inwards towards that center and be perpendicular to #v_T#.

You could also somewhat argue that in order for a circle to be made, the tangential velocity has to change direction through the sum of centripetal forces such that it always curves to form a circle below its tangential direction when the UCM is counterclockwise.

Then one must push inwards on the tangential velocity to force it to move in a circle, and thus the centripetal acceleration is inwards towards the center.