to make an example of someone to try to dissuade others from following in his footsteps

The killer is not as dangerous as the approval for his act. The government wants to prevent copycat vigilantes by making an example out of him. Regardless whether this guy is guilty or not, they'll drag him through the deepest mud and then string him up with the harshest possible sentence, to discourage others from gunning down CEOs.

I'm not adopting the conspiracy here, but if they can't find the shooter with the whole country getting behind him, then they would want the optics of finding the shooter. Which could backfire, of course. It would be very telling, for example, if the cops got real confident about it and then the real shooter made some kind of public display with the false shooter in custody.

They gain by preventing copycat killers. If other poors think that he got away with it, they might think they can, too.

Yeah, I get that. However, the optics of having a successful “martyr” symbol is very, very dangerous. A wide scope of narrative means difficult to control. Difficult to control introduces “motivator to action” symbols among a, I think, specific (and quite populous) demographic (think of all the young males with zero purpose, waiting to seize on an opportunity for a real life Mr Robot, for example)… “well if he could do it, get away with it, AND become a ’hero’, what’s stopping me from doing the same?”

Having someone, anyone, buys time to craft the narrative and gauge public sentiment and, most importantly, dampen the probability of a revolutionary ’spark’ if you will.

Obviously we don’t have enough information here. It very well could be the dude they have in custody. I am only sharing one possible theory based off my experience and observations. And there are a number of very suspect observations here that are in line with narrative management.