There is a class in between though. Those who can’t stop working and live on capital alone, but still have enough leeway to try and an asset that’ll improve their financial status. For example:

• Investing in higher education that can bring you higher salary. For the middle class it’s a gamble - maybe you won’t make it, or maybe you won’t be able to get a job that justifies your degree - but that’s categorically different from the rich who are pretty much guaranteed to graduate and get a good job using their connections (with the degree used as laundered merit) and from the poor who can’t afford to invest the time (let alone the money) because their families will be in big trouble for several years if they don’t work and bring income.
• Buying a house. Not a problem for the rich, not a possibility for the poor, but for the middle class it’s a huge thing - both in the effort it requires and the benefit of not having to rent (or being able to rent it to others)