Prison isn't just punishment, it's a deterrent to future criminals. If white collar criminals got to chill at home for few years, do you think we would have end up with more victims of fraud, or fewer?
It's not an effective deterrent because people who commit crimes don't often stop and think "oh if I do this I will get caught." In fact if you've ever reported crimes to police as a victim you've probably had the experience that there's really not much that they do other than take a police report and if it's a crime against property basically tell you to call your insurance company. Prison is a deterrent to you or me because we think things through before we act. But criminals often have really bad impulse control relative to the general population and so any preventative measures that rely on impulse control won't work on them.
There is more than one type of criminal in the world. What you’re describing is more true of the guys spontaneously grabbing purses or backpacks, not white collar criminals who spend years planning and maintaining their crimes. They’re far more likely to weigh the risk of serious enforcement, and jail is far more of a deterrent for them.
The US has a lot of people in prison for non-violent offences. It even has a huge number of people in pre-trial detention, where other countries have better bail systems and things like electronic tagging to allow people to stay out of jail before they are convicted.
I think you’d find a lot of support for the idea that we should have fewer people total in jail (i.e. funneling drug users into medical care, use community service for non-violent property crimes, etc.) but more white collar criminals going to jail.
As a simple example, I’ve heard a lot of people of all ages comment that the bankers would repeat the kinds of crimes they committed during the mortgage bubble because they made more profits than they were fined. Jail time is far more plausibly a deterrent in those cases.