That was also my belief up until this passage:

"More importantly, why did the Remembrall go off like that?" Harry said. "Does it mean I've been Obliviated?"

"That puzzles me as well," Professor McGonagall said slowly. "If it were that simple, I would think that the courts would use Remembralls, and they do not. I shall look into it, Mr. Potter."

In-universe, this is little evidence for or against anything. But from a narrative point of view, if the answer to "What did the Remembrall flare up about?" had been "Do not use the Time-Turner for showing off", this was the time to reveal it, rather than show Harry and McGonagall being confused. Certainly it wouldn't be an answer worth waiting over forty chapters for.