Straight lines don’t allow to add points to their centers / middles
So, I created a square, changed the line brush with something else, and then cut the 4 corners with scissor tool. Then, if I select the add anchor point tool, and click in the EXACT MIDDLE of the newly created line (which I cut from the square), it will not allow me to do so. It's just saying you only can add anchor point on the path line.
Please help me Adobe...

This is fixed in the latest Beta Build 29.5.0.15 — now straight lines allow to add a point at the center positions. Please check if this now behaves as expected and comment back.
However, the team didn’t make it into the snapping feature requested in comments — to be able to snap to middles of any segment, curved or straight. If you need one, please upvote this split request.
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Hans-Peter Banitz commented
In CorelDraw, the centres of paths are 'magnetic', so that, for example, guide lines can snap there. To find the centre in Illustrtor I have to add a point via script or AG PathScribe, which otherwise only inflates the document.