The illusion is actually measurable. For example when you cause activity in cortical motor neurons by direct stimulation, then the subject will move e.g. their arm involuntarily, but they will later convincingly explain that they've moved their arm on purpose because of an itch or because they just felt like moving their arm.

Think about it. Isn't there something curious about this (probably innate) way of representing actions that were initiated inside our brains?

There are more examples like these, e.g. in split-brain patients where the speaking side simply rationalizes whatever the non-speaking side does. All of that points toward the same idea: Our feeling of being an entity that can choose in each moment what to do next misrepresents what actually happens inside our brains. Under the hood, every thought and every movement can be tracked back to either noise or to a recurrent control program that in turn was shaped by a reward maximizing mechanism and by learned statistics of real-world dynamics. On top of that there is an episodic self-representation which basically tells a story about itself and which has a bias to represent autonomously moving objects such as the structure it is part of, by a free, self-initiated intentionality. That story it tells itself about itself can of course affect future actions, but again the way it does is fully contingent on the laws of nature, on genetics, on the individual life experience etc.

If someone stopped caring about their life after reading this because they are not in control of their thoughts and movements anyway, then even that is fully contingent on the experience of reading whatever I am writing right now and society might decide to contain said subject for everyone's safety (i.e. to ensure their continued flow of reward signals). This exposes the role of this innate feeling of free agency as a mechanism of behavioral control. We need this representation to be efficient at attributing certain outcomes to certain individuals so that we can correct their behavior.

But we don't need to give up on anything knowing this. We still can do blame attribution. Actually we can probably gain something by improving the representation of ourselves and agents in general: Everybody is deeply shaped by their individual experiences and often it is actually quite insightful to go on 'auto-pilot' and ignore our representational urge to be our own initiator for a while and just see what the recurrent circuitry in ours brain can come up with.