It's true it's moderate and push for gradual changes to ease everyone in, but it being appealing to more people doesn't make it centrist, I don't think. It's purpose isn't to balance in the center between the left and the right, but rather to use softer kind of force to move society left.
As in the example you used, what we consider right and (nowdays even far right) manages power without much fuss from the society, and is appealing to some despite it's facist undertones. Would you consider Republicans to be centrist? Because if you wouldn't, I'd argue that any democratic socialist party wouldn't be either.
I think the intent matters more than public opinion, you could sway the public with charismatic enough figurehead without changing anything about policy. I see the 'center' as more of the tendency not to change anything either way or balance between the 'extremes', and democratic socialism intends to be polite about beheading the capital class.
The popularity I've talked about referenced your point about it being moderate and easy on everyone nerves. I've oryginally started my previous comment by saying, that full blown socialism right here and right now would be good for everyone and it would be considered pretty leftist, but deleted that after deciding this part was pretty much obvious. Something being good doesn't make it centrist. That's why I stayed on point of public sentiment, which you seemingly invoked by defining center as moderate in the eyes of voters.
Say whatever you want about their hienous ideas, there wouldn't be an issue in USA right now or anxiety about Trump winning if they weren't reasonably popular. And I'm not conflating that popularity with doing good, but using their example to reject your argument about popularity making a political system 'centrist'.
I don't understand where did you get popularity = 'doing good' from me, but before we get into argument about that, I don't see how either of those would make a system centrist. 'Good' is relative, and further left would be 'better' by this logic, right? So how does that make a democratic socialism 'centrist' if 'doing good' is the measure you're using here? It being moderate is for the sake of popularity, gradual shift to the left so no one has any major complaints, and I think I've spoken enough about how I don't see popularity as reasonable measuring standard here.
Democratic socialism wants to overthrow the capitalism, bring socialism, give everyone free healthcare, have worker co-ops as default mode of working, UBI, yada, yada, all of those propositions are radical (as in fundamental) and definitely leftist. Instead of violent revolution this system proposes a reformative approach, and that's basicially the main difference from wide range of socialist systems that would attempt to implement the same things. So how is that centrist? Moderate, I get. Popular, sure. But center would refer to either a midpoint between the furthest right and left ideologies, or a minimal degree of change from the current political system, depending on how you want to define that word. I can't see Democratic Socialism fitting either of those definitions, so it has to be a leftist system. I don't see how it being moderate or popular would even influence that.