Setting aside all of the conspiracy theories, the main reason the DNR denies so many of the "mountain lion" sightings is that they simply weren't mountain lions. The DNR probably gets dozens of reports of mountain lions that turn out to be nothing more than bobcats or housecats for every legitimate mountain lion sighting. It doesn't take many mistaken reports or less than credible witnesses before the default position becomes skepticism and once that happens it takes really strong evidence to overcome it. I've heard enough stories from enough random people to have a pretty healthy skepticism myself and I've heard only a tiny fraction of the reports that I'm sure the DNR gets.
Now, I suspect there are other reasons that the DNR often downplays the handful of real sightings while also listing them as a protected species. I think that the way the Feds have tied the hands of the Western and Upper Great Lakes States to manage wolf populations because of their questionable listing as an Endangered Species combined with the push by some to consider eastern cougar populations as a separate and of course endangered population, likely plays a role in this reticence of the DNR to promote real mountain lion sightings. Neither the DNR, nor any outdoorsman, should want any part of the Feds hijacking the wildlife management of the state in order to protect "endangered" mountain lions. As long as the DNR can keep playing the infrequent legitimate sightings off as escaped pets and prevent a bunch of cats from being shot and ending up in the news, they can avoid Federal involvement.
I can kind of see your point about not wanting to encourage people to go out and start shooting them (if they exist) I guess. I really can't see how a bobcat (short stubby tail) could be mistaken for a cougar (long tail), and I really don't understand how a housecat (a really HUGE housecat would be 20 lbs.) could be mistaken for a 100 lb. cougar unless there had been a LOT of alcohol or possibly some hallucinogenic drugs involved, although I don't doubt these two scenarios do happen.