The character knock on Simmons is that he is a selfish stat-padder who doesn't care about winning and lacks competitive fire. I think there should be a legitimate concern that he is a soft player who will shy away from contact in the NBA.
Here are the comments on nbadraft.net for both players that relate to character:
Simmons - NBADraft.net Pros"Highly unselfish, team oriented player ... A supreme competitor and a proven winner, having captured three national high school championships (in three seasons) ... Attitude/character is by all accounts one of his greatest assets, as he is praised by many for being "a better person than player"
"Has tremendous strength, showing the ability to play right through contact and finish ... An elite level rebounder showing the grit and desire to grab every rebound in his area, and put in work on the offensive glass"
"He has the desire and work ethic to become a gym rat, and his shooting should improve considerably over time"
"He is a hardworking kid with the type of weaknesses, namely shooting, that can be improved upon ... Well spoken, confident kid who apparently has no trouble being a vocal leader, strong communicator ... Level headed, mature kid who seems to battle through adversity with ease, does not lose his cool or get down when things get challenging ... "
"The scary part is at only 19, he has already proven himself in one-on-one battles with the likes of Anthony Davis and James Harden (At Nike Academy), and one has to wonder just how much better he can become"
Simmons - NBADraft.net Cons"Can have scoring droughts in games related to both his confidence in his shot, and unselfishness ."
Ingram - NBADraft.net Pros"Has a high basketball IQ, good attention to detail ... "
Ingram - NBADraft.net Cons"Not a strong defender yet ... Doesn't possess great lateral quickness ... Often out of place on the defensive end ... His footwork when defending on the ball is subpar"
"Not an overly explosive athlete ... While he plays smoothly, can seem to be playing too slow at times or even come across as lazy ... Can develop into a better playmaker ... A shy, low key kid, should look to develop into more of a vocal leader ..."
Simmons - DraftExpress Cons- "Not very competitive defensively - takes plays off"
- "Questions about his character and overall approach"
Ingram - DraftExpress Cons- "Extremely talented but looks a bit too lackadaisical on the floor at times"
- "Overall effort and energy come and go, especially on the defensive end"
-"Does he have the mentality of a #1 option and top draft pick?"
Everybody throws criticism at Simmons for his attitude/character concerns, but people seem to be happy to completely ignore Ingram's character concerns and give him a free pass. Doesn't sound to me like his character is any better than Simmons at all.
It's not until you get down to Hield and Dunn (Bender, if you like him) that you really can step back and say "those are pretty high character guys".