rejected promise "Language client is not ready yet"

## Test Details

possibly correlated with https://github.com/aws/aws-toolkit-vscode/pull/6701 @KevinDing1

- Name of test: can't be sure, because uncaught promise rejection is async.
- OS of failure: linux
- VSCode version of failure (minimum/insider/stable): all
- Link to failing run: https://github.com/aws/aws-toolkit-vscode/commit/22d5a8a2f15bbf07ca43adff4ef8ef8ad3fabefb

## Log of Test Failure

```
rejected promise not handled within 1 second: Error: Language client is not ready yet
stack trace: Error: Language client is not ready yet
at LanguageClient.sendRequest (/codebuild/output/src2125471750/src/github.com/aws/aws-toolkit-vscode/node_modules/vscode-languageclient/lib/client.js:1789:19)
at LspClient.getIndexSequenceNumber (/codebuild/output/src2125471750/src/github.com/aws/aws-toolkit-vscode/packages/core/src/amazonq/lsp/lspClient.ts:214:50)
at async onAdd (/codebuild/output/src2125471750/src/github.com/aws/aws-toolkit-vscode/packages/core/src/amazonq/lsp/lspClient.ts:319:29)
Language client is not ready yet: Error: Language client is not ready yet
at LanguageClient.sendRequest (/codebuild/output/src2125471750/src/github.com/aws/aws-toolkit-vscode/node_modules/vscode-languageclient/lib/client.js:1789:19)
at LspClient.getIndexSequenceNumber (/codebuild/output/src2125471750/src/github.com/aws/aws-toolkit-vscode/packages/core/src/amazonq/lsp/lspClient.ts:214:50)
at async onAdd (/codebuild/output/src2125471750/src/github.com/aws/aws-toolkit-vscode/packages/core/src/amazonq/lsp/lspClient.ts:319:29)

```

## Additional Information

Start with packages/core/src/amazonq/lsp/lspClient.ts at getIndexSequenceNumber (line 214) and onAdd (line 319), then review the failure run and possibly related pull request 6701. Reproduce the Linux failure if possible and trace the language-client startup timing. Done means the asynchronous rejection is handled and the test run no longer reports “Language client is not ready yet” as an uncaught rejection.

Written by the indexing model from the issue text.