You made a nice observation, and it is implicit in the theory of the Poncelet Porism and the theory of conic pencils. It's a big topic - I won't offer any proofs - but hopefully I can point you to some references that will provide a starting point for understanding your observation. Along the way I'll point out some connections with previous observations you have made.
The textbook example of a conic pencil is the set of all conics passing through a given set of four points. A less discussed example is the set of all conics tangent to a given set of four lines, which we'll refer to as a tangential pencil. In a previous question you asked what the locus of the centers of a conic pencil would be - it turns out to be the so-called nine point conic. Now we can ask the same question for a tangential pencil. Surprisingly the centers are all on a straight line (the so-called Newton-Gauss line of the four tangential lines, apparently named after Newton because he discovered this fact).
There's a proof of this in Russell, Pure Geometry, pg 241.
But you might find it more fun to model this in Geogebra.
Why is this relevant? Because the three conics in your original question are members of the same tangential pencil. As a consequence, the centers are collinear.
Why are they in a tangential pencil? Now we have to get into the weeds of Poncelet's Theorem. A good but challenging reference is Poncelet's porism: a long story of renewed discoveries, I. Your diagram is echoed in Figure 30 of that paper:
The general Poncelet Porism setup takes $n$ points on a conic and then connects them to make a polygon which circumscribes a new conic (and by making more connections circumscribes more nested conics). These conics belong to a common conic pencil. Your construction intersects the tangents of the $n$ points and creates conics. This is the dual of the general construction, and the result is conics that belong to a tangential pencil (which is the dual of a normal pencil).
Fun fact: in an earlier question you asked about the conic passing through the eight points of contact of the four common tangents to two conics. This conic ($F=0$) is referenced in Figure 20 of the paper, and used in the analysis of Poncelet's Theorem.
Finally, check out the question Family of conics generated by two conics and my answer to that question, as well as the references in the answer.
One final remark. The conic pencils in Poncelet's Theorem have four base points, but they are imaginary. Similarly, the conics in the dual of Poncelet's Theorem (and the conics in your question) are tangent to four imaginary lines.
