Under capitalism each of these individual systems needs to turn a profit to be deemed worthwhile instead of treating the system as a whole and taking into account the economic externalities and benefits to the entire system.
There's no such thing as free things, there's just some people paying for other people's things, and surprise surprise some people don't want to use their hard earned money to pay for other people's transport and healthcare.
Giving more and more resources to those people does not make them magically more educated, productive, or congenial.
So it's not a "crabs in a bucket" mentality in that the greedy (which I assume means the wealthy here, as they're the ones funding the public system in the US specifically and the western world for the most part) are trying to keep the lower classes beneath them, it's that they are not interested in wasting their resources to no meaningful end other than increased consumption of low quality or worse goods and services.