You have captured much of the problem right here. Morphing the words of Jesus has been the, pardon me, lame excuse for backing perceived champions of a State-run welfare system. They now have more baggage than they bargained for.
Some people have a hard time differentiating between voluntary associations: a church is a group of people who have come together to form a church, and a country (in the American model) is a group of people who have come together to form a country. Since a church can (and should) collectively perform acts of charity, they see no reason a country cannot do the same.
This is a categorical mistake. Churches exist to express love; governments exist to express political power. Chairman Mao famously said, “Political power grows out of the barrel of a gun.” Love is not coercive. Therefore, acts of charity (love) cannot be performed by government. The same act performed by both church and state cannot not have the same moral effect: the loving act is voluntary, and the official act is an act of force.
Not all liberals are lazy and greedy jerks who want accolades for acts of charity performed by other people. Some liberals are merely kindhearted but confused.