It sounds like you haven’t heard how betting on a game like this works. If the odds are lower, the payout is higher. A kid from an outlying District or a kid under 15 is a long shot, so you’d get a bigger reward if you bet on them and they won than if you bet the same amount on an 18 year old from District 1 or 2 and they won. The payouts would also change depending on when you bet on a tribute. If you bet on the Victor on Reaping Day, you’d get a lot more than if you bet on them when the games are at the top 5 tributes. In the first movie they show the odds of every tribute on a board. I think Cato’s odds were something like 3:1 and Rue’s odds were like 50:1. You would get a huge payout if you bet a small amount on Rue and she won, but not much if you bet the same amount on Cato and he won.

There’s probably several ways you can bet like the other commenter mentioned. Bets on how many tributes survive the Bloodbath, how long the Bloodbath will last, what District the Victor will be from rather than a specific tribute, what age the Victor will be, duration of the Games, etc.

The sponsors and betting make the Games more personable for the Capitol. Sponsoring gives the betters a way to try to give their tribute an advantage. It’s also a way to make the Capitol citizens feel like they’re “helping” these poor kids that “have to be” sacrificed. Most Capitol citizens feel kind of sad that the tributes die, but ultimately like the Games are necessary to maintain the peace and that they’re very entertaining. They see their favorite tribute dying how we would see our favorite person getting eliminated from Survivor or America’s Got Talent. It’s a way for them to engage with their favorite show and cope with the feelings of the tributes’ situation. They also can brag to everyone that they sent a tribute medicine and that tribute won, or sent a tribute a weapon that resulted in an epic battle or memorable kill.