You should look into pricing the 4×4 concrete anchors. Four of those, and a few bags of concrete might be cheaper, considering lumber prices these days. It serves the added purpose of keeping the posts up out of the dirt.
Storms, and high winds are expected tonight. Looks like your project will see its first test. I hope you tamped those posts in good and solid. We'd hate to hear the wind got some catastrophic lift underneath it.
I probably would've used post anchors, set in concrete. But that would've been more about keeping the headroom, that's sacrificed by setting the posts in the ground.