examples: refactor error logging in error-handling example (#3527) #5825

Job	Run time
check-docs	34s
cargo-hack	2m 45s
check	37s
cargo-public-api-crates (axum-core)	19s
cargo-public-api-crates (axum)	35s
cargo-public-api-crates (axum-macros)	22s
cargo-public-api-crates (axum-extra)	28s
dependencies-are-sorted	5s
Spell Check with Typos	6s
cargo-deny check (advisories)	23s
cargo-deny check (bans licenses sources)	28s
test-nightly	1m 6s
test-versions (beta)	1m 11s
test-versions (stable)	1m 18s
test-msrv	2m 4s
armv5te-unknown-linux-musleabi	37s
test-docs	1m 0s
wasm32-unknown-unknown	18s
	14m 16s

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