You should really just instrument your application with metrics for a time series database like Prometheus, Influx, or OpenTSDB. You'll find a mature set of tools and methodology. Server Timing looks very naive.
Application instrumentation - whether via Prometheus, StatsD, Scout, New Relic - solves a very different problem than this. The server timing metrics here are actually extracted from an APM tool (Scout), so you get the best of both worlds.
With those tools, you do not get immediate feedback on the timing breakdown of a web request. At worst, the metrics are heavily aggregated. At best, you'll need to wait a couple of minutes for a trace.
Profiling tools that give immediate feedback on server-side production performance have their place, just like those that collect and aggregate metrics over time.
