refactor error logging in error-handling example #5816

Job	Run time
cargo-hack	3m 11s
cargo-public-api-crates (axum-core)	41s
cargo-public-api-crates (axum-macros)	40s
cargo-deny check (advisories)	34s
cargo-deny check (bans licenses sources)	27s
cargo-public-api-crates (axum)	55s
cargo-public-api-crates (axum-extra)	47s
dependencies-are-sorted	5s
Spell Check with Typos	9s
check	44s
check-docs	31s
test-nightly	1m 5s
test-versions (stable)	1m 9s
test-msrv	2m 54s
test-docs	1m 1s
test-versions (beta)	1m 14s
wasm32-unknown-unknown	27s
armv5te-unknown-linux-musleabi	30s
	17m 4s

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