But he's not proclaiming himself a democrat. He keeps a bit of distance for himself from democracy and the democrat, and he gives the politicians of his day a reason for not going the socialist route: "The germinal conceptions of democracy are as free from all thought of a limitation of the public authority as are the corresponding conceptions of socialism; the individual rights which the democracy of our own century has actually observed, were suggested to it by a political Philosophy radically individualistic, but not necessarily democratic."
To me, Woodrow Wilson comes across as somebody on the fence here. Is he going to stay with those individual rights concepts that keep politicians from embracing socialism or his he going to embrace "democracy" and with it socialism? It looks like he's getting ready to take that plunge but hasn't jumped yet. In the same way, he's rejecting "sentimentality" about the poor and preparing to embrace a less emotional creed of "policy" and "administration" but he's a little timid about making the break.
It's also worth noting that he identifies socialism with equality of opportunity, "every man shall have an equal chance with every other man," not with equality of condition. That's a spin most socialists wouldn't give their ideal. In any case, it looks to me like he's struggling to give birth to something and he's not really sure yet how to present his child to the public. To change the metaphor, he's almost ready to go out in a socialist suit, but he's still aware that it might make him an unwelcome spectacle.
For the larger question of socialism and democracy, you could say just about any government is going to interfere in the economy. All the more so if it's elected by universal suffrage and its officials want to be reelected. A republic with a more restricted franchise might refrain from "socialistic" measures (though it might also intervene in the economy in the interest of its own constituency), but such regimes don't tend to last in democratic eras. We got democracy with Andrew Jackson, though it didn't bring those socialistic measures for another century. It's doubtful we could go back to the America of Hamilton and Jefferson.
You mention a lot of interesting points about how socialism and democracy can (but do not necessarily have to) be coextensive. The problem has been that a Constitutional Republic has been incompatible with purer forms of Democracy. There are a host of things today that are unconstitutional if only a slightly literal interpretation is granted. When one assumes merely that the words of the Constitution have meanings common to the understanding of the Founders, then it is clear that we no longer are ruled by the Constitution of our Founders.
I often ask my liberal friends “What limits does the Constitution places on the ability of one group to take property from another through taxes?” Or I might ask, “How are the wealthy protected constitutionally from the ability of anyone to seize and redistribute their property on a national-scale?” They honestly have never even thought of such a question. Their minds are constantly geared towards how to take more. Why would anyone want there to be a constitutional-limitation?
The Founders did have a limitation. It was the Consitution itself with the notion of delegated powers. This is where Jefferson was right and Hamilton for all his financial acumen was wrong. Implied powers allow the General Welfare clause and Interstate Commerce clause to obliterate any notion of federal restraint in realizing the will of a majority on a minority...and that most importantly involves a minority with accumulated wealth. Hamilton might be horrified to see where his philosophical tendencies have lead. I certainly do not think he was a socialist. He did err far too often on the side of statism, and that way of thinking is what allows liberals to justify their ever-expanding policies.