[Bug]: `RenderDocument` introduces `page.close()` hang

@dgozman is already working on this.

Since Jul 31, 2026.

### Version

ToT

### Steps to reproduce

```ts
test('should close page while reload is committing', async ({ context, browserName }) => {
test.skip(browserName !== 'chromium');
for (let i = 0; i < 20; i++) {
const page = await context.newPage();
await expect(page.evaluate(() => {
location.reload();
return new Promise(() => {});
})).rejects.toThrow(/navigation/);
await page.close(); // intermittently hangs
}
});
```

### Expected behavior

The test passes fine.

### Actual behavior

`page.close()` hangs. To expose this behaviour, it's important to have the loop inside the test, running it without the loop but with `--repeat-each` doesn't expose it. I suppose it's linked to races within the context or something.

### Additional context

This started at https://github.com/microsoft/playwright/pull/41303.

Discovered through https://github.com/microsoft/playwright-python/pull/3161.

### Environment

```shell
Simon's Mac
```