https://chromium-review.googlesource.com/c/chromium/src/+/1189711 records some metrics during ash::Shell teardown. I'm sure there are other performance metrics we record during shutdown elsewhere.

Chrome's MetricsService outlives ash::Shell.

~BrowserProcessImpl() destroys metrics_services_manager_ (ultimately owns MetricsService)
ChromeBrowserMainExtraPartsAsh::PostMainMessageLoopRun() destroys ash_shell_init_ (owns ash::Shell)

There's also this call in chrome_browser_main.cc:

    browser_process_->metrics_service()->Stop();

https://cs.chromium.org/chromium/src/chrome/browser/chrome_browser_main.cc?type=cs&sq=package:chromium&g=0&l=2108

However, under multi-process-mash we may have a shutdown race. We need to make sure shutdown metrics in ash get recorded, or at least persisted for upload for the next session.