This fixes an issue that there were sticky launch errors. When the
[following code
path](a15f0f301b/src/context.ts (L307-L339))
was throwing, the Error was stored in the Promise and not cleared
afterwards, this meant:
- If a browser was not there and the user tried to install it via
`browser_install` it was never working since the error was sticky.
- If other errors like CDP is not available yet etc. error appear a
re-connect would not work - the MCP server would require a restart.
Test plan: Since we don't have any `browser_install` tests I added a CDP
test for now to cover this bug.
