But they're not exactly fans of modern democracy, either, so it seems that foundational-axiomatic positions do often lead to uncompromising absolutism of one kind or another.
Really? That's a very strong statement which I would like to see to be backed up with some axioms. :-) (Because both analytic philosophy and democracy seem more preferable to me.)
Anti-foundationalists, in contrast, think that reality is far too complicated for a single person to figure out, and so we need extended democratic discussion. They also tend to believe that average citizens have a considerable degree of rationality and good motivation, and so can be trusted to be responsible voters. This is, of course, imperfect, but as Churchill famously said, it is better than all the alternatives.
As far as analytic philosophy goes, the classic version is foundationalist-axiomatic, but later versions, such as Strawson, are anti-foundationalist. Later Wittgenstein I think is a mixed case.
With Heidegger a notable exception.
The contrary view is that we can know about reality, but our understandings are always imperfect, including not being able to express everything in a single set of precise concepts. In my view this was Aristotle's overall view, but in any case in the last century and a half, Western philosophy has swung in this direction, including Whitehead, Heidegger, the later Wittgenstein, and all the Pragmatists. Putnam was a leading American philosopher in this anti-foundationalist trend.
Let me add that the foundational-axiomatic position tends to imply that political rule should be by something like a philosopher-king, while I think the anti-foundational one implies liberal democracy.