Use `Result` for `process_analysis_request`. #2883

Job	Run time
Linux x64 - Build - Profile 'debug'	1m 16s
Linux x64 - Test	1m 34s
Linux aarch64 - Build - Profile 'debug'	1m 20s
Linux aarch64 - Test	1m 33s
macOS x64 - Build - Profile 'debug'	3m 32s
macOS x64 - Test	2m 35s
macOS aarch64 - Build - Profile 'debug'	1m 43s
macOS aarch64 - Test	1m 43s
Windows x64 - Build - Profile 'debug'	6m 25s
Windows x64 - Test	3m 23s
Clippy	2m 3s
Rustfmt	1m 6s
	28m 13s

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