refactor error logging in error-handling example #5817

Job	Run time
check-docs	18s
check	39s
cargo-hack	2m 44s
cargo-public-api-crates (axum)	55s
cargo-public-api-crates (axum-core)	35s
cargo-public-api-crates (axum-extra)	51s
cargo-public-api-crates (axum-macros)	40s
dependencies-are-sorted	4s
cargo-deny check (advisories)	24s
cargo-deny check (bans licenses sources)	31s
Spell Check with Typos	7s
test-nightly	1m 1s
test-versions (beta)	1m 10s
test-docs	1m 7s
test-msrv	2m 40s
armv5te-unknown-linux-musleabi	51s
test-versions (stable)	1m 49s
wasm32-unknown-unknown	18s
	16m 44s

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