A social philosophy could well be based on truths about human nature, if there are any. I think there are : we have limited knowledge, limited rationality, we have shared vulnerabilities to one another. This list looks sound to me and could probably be extended.
Far more doubtful in my view is a social philosophy based on moral or other axiological truths. Even if there are such truths, there is widespread and intractable disagreement about what they are. This severely curtails their practical relevance even if they exist.
However, there can be objectivity without truth, Rational interpersonal agreement can take its place. Consider a very simple (and simplified) example.
Suppose we have a society of just two people, X and Y. The point I'm going to make is unaffected by the numbers involved.
X has equally strong preferences for A, B, C, D, E (these can be institutions, practices, goods and services or whatever).
Y has equally strong preferences for A, B, C, F, G.
None of these preferences can be fulfilled without the joint co-operation of X and Y.
Resources allow for only three preferences to be fulfilled. Isn't it rational for X and Y to agree on a social philosophy that endorses and fulfils A, B, and C, which they share, rather than to abort discussion because neither can have all they want - with the result, in face of the need for joint co-operation, that neither X nor Y gets anything they want ?
Truth doesn't come into the picture - moral or axiological truth.
This kind of approach to objectivity through rationality rather than truth is clear (and far more elaborately developed) in the ethical theory of Kant (1724-1804) and in the social philosophy of John Rawls in A Theory of Justice, 1971.