refactor error logging in error-handling example #5818

Job	Run time
cargo-hack	3m 1s
check-docs	32s
check	41s
cargo-public-api-crates (axum-core)	40s
cargo-public-api-crates (axum-macros)	36s
cargo-public-api-crates (axum)	57s
cargo-public-api-crates (axum-extra)	46s
cargo-deny check (advisories)	30s
dependencies-are-sorted	6s
Spell Check with Typos	7s
cargo-deny check (bans licenses sources)	24s
test-versions (beta)	1m 6s
test-versions (stable)	1m 18s
test-nightly	1m 5s
armv5te-unknown-linux-musleabi	30s
test-docs	1m 0s
test-msrv	2m 42s
wasm32-unknown-unknown	19s
	16m 20s

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