Defer DevTools structured metrics logging initialization.

This change moves the call to `MaybeStartLogging()` from the `DevToolsUIBindings` constructor to `SetDelegate` and to the each `Record*` method. This ensures that SessionStart will have it's parameters  populated values from the delegate.

In case we stay with default delegate, (rare mostly devtools:// URLs in a regular tab), initialize in `Record*` methods.

Bug: 420837547
Change-Id: I87b6d555a587997cd4247331db2ec3dab6c0324d
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/7006491
Reviewed-by: Andrey Kosyakov <caseq@chromium.org>
Commit-Queue: Andrey Kosyakov <caseq@chromium.org>
Auto-Submit: Danil Somsikov <dsv@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1524403}