If the Constitution only allowed taxes where everyone pays the same amount, how much socialism would we really want?

This isn't a "gotcha" question, I genuinly wonder about this myself: If government programs were constitutionally forbidden to do "wealth redistribution", how much socialism would I still want? Now, I looked at the state expences of Germany, and found that actually most of what I see isn't really redistribution. It's things like a public health and unemployment insurance, public pensions, the military, infrastructure, child benefits, police etc. That is basically all something that I'd want to keep funding whether we let the rich pay much of it through a progressive tax or whether everyone pays into it equally through an equal amount tax. 

Now, when we come to college, which is free in Germany, the question becomes a bit more tricky. For one, it is obvious that making college tuition-free means more people end up graduating, which boosts productivity. On the other hand, why should a baker pay for some soon-to-be rich doctor's university degree, if the doctor doesn't pay a dime more back afterwards than the baker pays into it. Ideally, in this situation, I'd probably still run state-owned universities, but only charge people beyond a certain income to attend it. Except in a way that's also wealth redistribution, so if that iss also forbidden, I could imagine that I would just let the public colleges charge as small a tuition as possible, to just barely cover their own expences and that's it. Public colleges are still a great thing - because they don't have an incentive to cut costs everywhere as private enterprises often do, or be overly lavish -  but if that means the lower working class has to become even poorer, then I think that's where society sharing the cost should be done.

I think many people aren't really aware that socialism only means the state controls a means of production, not that it can't charge people to use it. In many countries, water and electricity distribution is controlled by the government, which doesn't mean water is free, though, so you still have the incentive to not waste water. It's just that then you can be fairly certain that you're not screwed over by some company that has a monopoly on water supply in your region, so you're forced to take their prices. And no, cutting regulations will still not take away the huge costs of getting started as a water company, so competition would remain extremely low.

In summary: We as leftists shouldn't look at issues with the mindset of "How can I tax-and-spend here, to solve this problem?", just as right-wingers shouldn't look at every issue and say "Oh, look, something's not working. I'm sure if we stop doing anything collectively it will suddenly all start working fine". Instead, we should be aware that taking a country right or left just for the sake of taking the country right or left is never the solution. We should take political ideologies as a useful tool to solve issues, and take careful deregulation, regulation, tax-and-spend, tax cuts, tax credits, etc, we should take all of that as a serious possibility, examine for each case individually what its effects would be, and then choose the best one. That's how we can make progress, not by buying into that notion that one idea, whether it's state control or lack thereof, will suddenly turn everything into flowery rainbow puppies jumping over green fields, and we'll all end up holding hands and singing Kumbaya until the end of time. Reality is a bit more complicated than that