refactor error logging in error-handling example #5822

Job	Run time
Spell Check with Typos	8s
check-docs	42s
cargo-hack	2m 57s
cargo-public-api-crates (axum-core)	17s
cargo-public-api-crates (axum-macros)	16s
cargo-public-api-crates (axum)	37s
cargo-deny check (bans licenses sources)	29s
cargo-deny check (advisories)	29s
check	40s
dependencies-are-sorted	5s
cargo-public-api-crates (axum-extra)	28s
test-nightly	1m 2s
test-versions (stable)	1m 41s
test-msrv	2m 1s
test-docs	1m 0s
test-versions (beta)	1m 11s
armv5te-unknown-linux-musleabi	32s
wasm32-unknown-unknown	20s
	14m 55s

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