fix(react): Prevent custom page remounts in profile components #29020

Job	Run time
Check Permissions	6s
Formatting \| Dedupe \| Changeset	52s
Build Packages	37s
Unit Tests (**)	1m 10s
Unit Tests (**)	1m 15s
Publish with pkg-pr-new	1m 5s
Static analysis	1m 2s
Integration Tests (localhost, chrome)	58s
Integration Tests (ap-flows, chrome)	1m 3s
Integration Tests (fastify, chrome)	57s
Integration Tests (generic, chrome)	1m 30s
Integration Tests (handshake, chrome)	55s
Integration Tests (astro, chrome)	1m 2s
Integration Tests (express, chrome)	58s
Integration Tests (handshake:staging, chrome)	1m 26s
Integration Tests (sessions, chrome)	1m 0s
Integration Tests (nextjs, chrome, 15)	1m 0s
Integration Tests (quickstart, chrome, 16)	56s
Integration Tests (chrome-extension, chrome)	1m 47s
Integration Tests (tanstack-react-start, chrome)	59s
Integration Tests (react-router, chrome)	1m 5s
Integration Tests (sessions:staging, chrome)	1m 4s
Integration Tests (nuxt, chrome)	1m 21s
Integration Tests (billing, chrome)	1m 2s
Integration Tests (machine, chrome)	1m 0s
Integration Tests (quickstart, chrome, 15)	1m 10s
Integration Tests (nextjs, chrome, 16)	1m 10s
Integration Tests (cache-components, chrome, 16)	2m 23s
Integration Tests (hono, chrome)	1m 0s
Integration Tests (vue, chrome)	58s
Integration Tests (custom, chrome)	57s
	33m 48s

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