During his eight years in office, Obama deported a record number of immigrants, which is traditionally a Republican thing to do. He can’t openly take credit for it, because it would alienate his base — and if he tried, Republican voters literally would not believe it. Part of that is the right-wing echo chamber, but part of it is also that people reasonably expect the liberal party to do liberal things.
Obamacare is another great example: it IS the market-based alternative to socialized medicine, which is traditionally a Republican thing. But he can only sell it to his base as a necessary compromise (despite the fact that it passed solely with Democratic votes), and meanwhile Republican voters still think it’s socialized medicine — because, again, they expect the liberal party to do liberal things. In this example, we have the added twist that they assume anything the liberal party does is liberal, hence the health care debate is now skewed sharply to the right as a Republican policy becomes the far left edge of possible options.
We can see the same dynamic with gun control. Democrats basically decided to give up on this issue and haven’t pushed any serious gun control measures in a long time, other than symbolic gestures after particularly horrifying mass shootings. But the gun lobby refuses to take yes for an answer: they still rile up their base with images of Obama or Hillary sending in the jackbooted thugs to take all the guns. Yet again, we’re dealing with the self-enclosed fantasy world of the right, but also with the fact that people reasonably assume that both sides of a controversial and important issue will be represented in the political system.
In these and so many other cases, centrism is a clear political loser — you turn off your own supporters and gain nothing. If you were designing a political strategy with the goal of long-term defeat, I don’t think you could do better than actual existing Democrats.