📈 Add telemetry metric for session manager init duration

[thomas-lebeau]

Motivation

The session manager is now async. We want to measure how long initialization takes in the wild to understand its impact on SDK startup time.

Changes

• Add a new SESSION_MANAGER_INIT_METRICS_TELEMETRY_NAME telemetry metric
• Measure elapsed time from the start of startSessionManager() to successful initialization using relativeNow() (performance.now())
• Metric is emitted for both RUM and Logs (since both call startSessionManager)
• Only emitted on successful initialization (early returns for missing strategy, stopped thread, or revoked consent are skipped)
• Includes a monitor-until: 2026-10-15 comment

Test instructions

• Existing session manager unit tests pass (54/54)
• Typecheck passes

Checklist

• Tested locally
• Tested on staging
• Added unit tests for this change.
• Added e2e/integration tests for this change.
• Updated documentation and/or relevant AGENTS.md file

[cit-pr-commenter-54b7da]

Bundles Sizes Evolution

📦 Bundle Name	Base Size	Local Size	𝚫	𝚫%	Status
Rum	168.22 KiB	168.31 KiB	+98 B	+0.06%	✅
Rum Profiler	6.09 KiB	6.09 KiB	0 B	0.00%	✅
Rum Recorder	27.10 KiB	27.10 KiB	0 B	0.00%	✅
Logs	53.80 KiB	53.97 KiB	+177 B	+0.32%	✅
Rum Slim	126.58 KiB	126.65 KiB	+74 B	+0.06%	✅
Worker	22.99 KiB	22.99 KiB	0 B	0.00%	✅

🚀 CPU Performance

Action Name	Base CPU Time (ms)	Local CPU Time (ms)	𝚫%
RUM - add global context	0.0163	0.0045	-72.39%
RUM - add action	0.0451	0.0172	-61.86%
RUM - add error	0.0554	0.0165	-70.22%
RUM - add timing	0.0083	0.003	-63.86%
RUM - start view	0.0588	0.0141	-76.02%
RUM - start/stop session replay recording	0.0023	0.0008	-65.22%
Logs - log message	0.0578	0.018	-68.86%

🧠 Memory Performance

Action Name	Base Memory Consumption	Local Memory Consumption	𝚫
RUM - add action	64.32 KiB	64.50 KiB	+178 B
RUM - add global context	38.44 KiB	37.79 KiB	-667 B
RUM - add timing	42.09 KiB	41.51 KiB	-592 B
RUM - add error	69.03 KiB	68.57 KiB	-474 B
RUM - start/stop session replay recording	41.75 KiB	41.12 KiB	-643 B
RUM - start view	477.81 KiB	476.46 KiB	-1.35 KiB
Logs - log message	53.80 KiB	53.65 KiB	-161 B

🔗 RealWorld

[datadog-prod-us1-6]

✅ Tests

🎉 All green!

❄️ No new flaky tests detected
🧪 All tests passed

🎯 Code Coverage (details)
• Patch Coverage: 100.00%
• Overall Coverage: 77.51% (+0.01%)

_{This comment will be updated automatically if new data arrives.
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