That is what I also think: having infinite money means money is never a problem, and the reason why gamblers place bets is for the money. I know the fun can also be there, but if money were not involved in gambling, I know many gamblers wouldn't be into it.
If I had infinite money, I also would not gamble; I would only enjoy my money and look for other games that I don't need to spend money on. Gambling is a game of loss and win; I wouldn't want to lose money in gambling when I already have so much. Indeed, gambling is a business. People are more interested in it because of the money, but they always pretend as if the fun in gambling is the most interesting thing about it.
Categories of games in gambling can differ greatly from one another. Not all games that involve financial gain or loss are solely focused on money. While I agree with you that the primary goal of gambling is money, I also know that people play for the thrill and excitement. For example, a person with an unlimited supply of money may still be passionate about poker, because for him it's a way of claiming himself, and the financial win or loss in such a game serves as proof of his skills and abilities.