Simplify `TestClient` implementation (#3096) #5144

Job	Run time
check-docs	27s
cargo-hack	5m 13s
check	1m 32s
cargo-public-api-crates (axum)	28s
cargo-public-api-crates (axum-core)	19s
cargo-deny check (advisories)	23s
dependencies-are-sorted	13s
cargo-public-api-crates (axum-extra)	30s
Spell Check with Typos	5s
cargo-deny check (bans licenses sources)	23s
cargo-public-api-crates (axum-macros)	17s
test-nightly	1m 26s
test-docs	1m 37s
armv5te-unknown-linux-musleabi	42s
test-msrv	2m 45s
test-versions (stable)	2m 52s
wasm32-unknown-unknown	16s
test-versions (beta)	3m 1s
	22m 29s

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