I’m not sure why they would reduce party influence either. Features like being robust against spoilers would seem to most benefit major party candidates.
For majors the benefit is clear: you don't get spoiled.
For minors, you don't have to overcome the barrier of being a spoiler if the race looks remotely close.
And of course voters are not disadvantaged just for having two acceptable choices rather than one.
There only functional difference between it and say, the original French two-round system that Maurice Duverger (of Duverger's law) contrasted with FPTP is that someone who wins an outright majority in the first election (an open "primary" in California) is not immediately elected.
The fact the second round is FPTP doesn't change the overall voting system. With only two candidates for a single seat, most voting systems degenerate to FPTP, but none of the issues related to FPTP are present either (there are no clones, no strategic voting, etc.)