Yes, people do that, I think that's the whole point of gambling is, right? You risk your money in the hopes that it's going to double in value which gives you profit. Sure you could do that and not bet but that's going to only be the peak, with betting, you're bound to get more excited and saddened when you're watching a game depending on the result, winning a bet on a sports that you've watched and loved is a different kind of fun if you ask me, plus to many people, betting is just a second thing to them, they just bet for the sake of they have something to do other than just watching.
Well, as much as I agree with most of the things you have said, let me say that, I personally; sometimes prefer to bet on a game than watching it, while in some games as well, I will prefer to just watch and not bet.
So, in other words, lets just say that betting for what ever sake, simply boils down to the gambler who is playing, we all are different in our different ways after all.
And going back to the main discussion, it is true that someone watching a match which he or she had a bet on, gives a different kind of feeling that watching a game that you don't have any bet on it.
I remember one time I was watching a game between and Arsenal and Tottenham, I watched the way Arsenal was playing and felt that they may likely win the match, I immediately picked up my phone, logged in on my casino account and placed a bet on Arsenal, and immediately I did, I noticed that my entire body system changed, I began to have this navious or anxious feelings when ever Tottenham had the ball and was heading towards arsenal's goal post in attemptds to score.
So, betting on a game, is another fun way of enjoying a match, and not just by watching it alone.