Christ does exhort his followers to render unto Caesar what is Caesar’s, and Democrats believe the state acts on behalf of the populace to perform tasks that we individually could not do. So a community can clothe and house the poor better than an individual can.

Clothing and housing are artefacts of this world, not the kingdom of Christ. Ergo rendering unto Caesar what is caesar’s involves providing clothing and housing to those who do not have it, with the state as proxy and effort multiplier.

Yeah, the "render unto Caesar" episode was so much deeper than just "pay your taxes."

Also, whatever the state does, it accomplishes through the threat of violence to collect taxes. For a Christian, the ends are as important as the means.