I do not have a reference to back my claims. Below is just my analysis of the stories I've read or watched about Rome and the Gladiator concept.
Gladiators were slaves(war soldiers or people condemned to punishment in their lands). They could choose (debatable) to be trained as Gladiator and fight to death in the arena or work their way to death in mines and other such hard labor. When you have death on both sides, you will tend to choose a glorious one.
If you go by this narrative, Gladiator games were gory but not barbaric
Barbaric would be culling the men in cold blood. The Gladiator had a ray of hope that one day, if they pleased the crowds and kings beyond measure, they can hope to gain freedom. If not, they were already living a better life then they were condemned too.
If I was a Roman citizen at that time, with the widest perspective, I would be repulsed by the concept of slavery, but with a normal perspective, shared by
most of the common people, I would welcome the games because of the above perspective.
So I dont think a consensus would be achieved from the common people to moot out the games. If somebody from position of power showed them the wider perspective of slavery, then yes, common people will gather to stop the games. Another perspective is that these games were portrayed as a means to please the Gods,
specifically the offerings of blood. This perspective was shared by every Roman & faith is irrational. So how could they oppose the games from this perspective?
Coming to Hunger Games, Districts rebelled against the Capitol and hence were punished through Hunger Games as a reminder. But the tributes were never slaughtered. It was gory but not barbaric (applying previous logic). The common people of the Capitol (assuming that they forgave the districts for their rebellion on the perspective of it being a fight for their freedom and independence rather than a rebellion to annihilate the Capitol), with a normal perspective, would view the games in the same sense as Gladiatorial games. The people of the other districts (slaves of the Capitol) were condemned to a hostile journey to their inevitable death. Here is a chance where one from all the 12 districts could stand alone in the arena and finally live a life of
free(ish)dom with ample food and aesthetic pleasures.
Mind you, people from the higher districts trained their tributes for the games (career tributes). So some districts viewed it as games and a way to amend the ties with the Capitol. That was the case before Katnip was the winner. The people of the Capitol were so impressed with her and Pita and their relationship of love that they demanded and cherished her freedom (in the sense of a Gladiator being
granted freedom in the arena). She was the one person the Capitol people never wanted harmed. She became an object of admiration and affection to the people of the Capitol.
Talking about the people from 12 districts, she became a "beacon of hope" for them. The one they could rally around. Since the question is about the Capitol people this narrative doesn't matter.
When she was re-selected for the Games, many people would have been heart broken. But there was always a sense of hope that she would stand alone again. But the news of her pregnancy was the blow that knocked the wind of out them. She would never survive and would die with an unborn child.
So, if I am a Capitol citizen, I would view Hunger Games as a means for entertainment, retribution and a means to secure a better future for one of the condemned ones from Districts, a silver lining. But never barbaric.
Considering a wider perspective of slavery, will I vote down the games in the times of Monarchy? NO, and I will live out my lifetime with security and
a pang of guilt. YES and I will struggle my (and my family's) way towards a barbaric death with a ray of hope for a better future for all.
Most people would choose NO. Legendary few will choose YES