Indeed, you are misunderstanding the comparison.
The comparison is between the very similar categories of laws discussed in the two chapters. Both chapters are uniquely about whether the laws of carrying on Shabbos apply to wearing various kinds of things; the fifth chapter is about what animals might "wear" (bridles, collars, saddles, bells, etc.), and ...
The commentaries I've come across all seem to agree that Zerubavel was indeed the biological son of Pedaya.
A number of approaches are offered to explain why he is nonetheless referred to as the son of Shealtiel:
Radak to Divrei HaYamim I 3:18-19 understands that Pedaya was a son of Shealtiel, such that Zerubavel was a grandson of Shealtiel. Zerubavel is ...
In summation it does not mean that the Mashiach will save Bnai Yisrael, but that Hashem will save the Mashiach. See the references below.
Chavakuk 3:13 shows the JPS translation as
You went forth to rescue Your people, to rescue Your anointed. You
have crushed the head of the house of the wicked, uncovering it from
the foundation to the neck-forever.