You might be able to put away some money while paying rent even though you don't have any equity in your rent.

How does this figure? The only case where this would be a thing is if your rental payment is less than a mortgage payment would be for that property, which (especially these days) is not typical. It doesn't even make sense when you think about it: the landlord needs to pay their mortgage, plus costs for maintenance, plus profit. The primary barrier to buying is down payment and closing costs, which is why renting is only better if you're moving around a lot (more than once every 5 years or so).