There are approaches to real analysis that use an axiomatization of nilpotent infinitesimals to enable rigorous synthetic reasoning about infinitesimals, which is arguably closer to the reasoning employed by mathematicians prior to the arithmetization of analysis.

While I am far from an expert, it seems that the conventional narrative is that increasing doubts about the validity of some of the results of classical algebraic geometry led to a similar arithmetization of algebraic geometry by Zariski and Weil. The gap between the resulting methods and the underlying geometric reasoning seems a bit wider than the gap between $\epsilon$-$\delta$ arguments and infinitesimals, although that is entirely personal opinion and may only be due to my familiarity with the latter.

Is it possible to do algebraic geometry in a synthetic manner that enables rigorous reasoning but is closer to the style of argument employed by classical algebraic geometers?