You are correct---in your system, someone can be surrounded by a society that is fundamentally positive because no one is wanting for basic needs, and leverage that fact to be wealthy because they don't have to worry about a certain class of problems, yet not contribute back into the system that has allowed their wealth to accumulate. This is as much a choice as a welfare recipient who could work, but is choosing not to. Arguably, that society isn't really possible, though, since given the option of not contributing back into the system, there is a high probability that those who accumulate wealth would choose not to do so, especially if it's inherited, and so even if you could trip into a society that met basic needs for everyone through the wealthy choosing to contribute, it would likely vanish in the span of a generation or two. Evidence for this being that we've had societies where the wealthy could choose not to contribute to their society for millenia, and meeting basic needs is something we've only started brushing at in the last century.
By the way, no, the goal here is not to spend the money as efficiently as possible. The primary goal is to succeed at something that is known to improve society as a whole and, perhaps, will allow that society to produce goods better and more effectively. The secondary goal is to do so as efficiently as possible. This is why you don't find much support in countries with nationalized healthcare for abolishing nationalized healthcare; instead, the support is for reforming it or making it more efficient or less corrupt or what have you. The primary goal met, it's on to the secondary goal. Before government was used as a vehicle for achieving many of these things, they simply weren't an option for most.
Government is where we come together as a people and say "yes, this is the cost that we have decided is worth paying in order to have the society that we want". That encompasses both sides---taxes ("what is worth paying") and policy ("how we achieve the society that we want"). Let's grant that there might be less competition and less efficiency (arguable), but there is greater universality, and I would say there are goals where that universality is far more important than the tradeoffs.