refactor error logging in error-handling example #5824

Job	Run time
check-docs	59s
cargo-hack	2m 45s
check	42s
dependencies-are-sorted	4s
cargo-public-api-crates (axum-macros)	14s
cargo-public-api-crates (axum)	36s
cargo-public-api-crates (axum-extra)	32s
Spell Check with Typos	6s
cargo-deny check (bans licenses sources)	28s
cargo-public-api-crates (axum-core)	18s
cargo-deny check (advisories)	32s
test-nightly	1m 12s
test-versions (beta)	1m 14s
test-versions (stable)	1m 14s
wasm32-unknown-unknown	21s
test-docs	56s
test-msrv	2m 4s
armv5te-unknown-linux-musleabi	27s
	14m 44s

Provide feedback