Simplify `TestClient` implementation #5143

Job	Run time
cargo-hack	6m 0s
check	1m 42s
check-docs	22s
cargo-public-api-crates (axum)	32s
cargo-deny check (advisories)	22s
cargo-public-api-crates (axum-core)	25s
dependencies-are-sorted	14s
Spell Check with Typos	6s
cargo-deny check (bans licenses sources)	23s
cargo-public-api-crates (axum-extra)	35s
cargo-public-api-crates (axum-macros)	21s
test-msrv	2m 27s
test-docs	1m 46s
armv5te-unknown-linux-musleabi	46s
test-nightly	1m 38s
test-versions (stable)	2m 52s
wasm32-unknown-unknown	16s
test-versions (beta)	2m 52s
	23m 39s

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