create-react-app v3 doesn't show TypeScript errors in overlay

🐛 bug report

Description of the problem

TypeScript errors in a create-react-app sandbox are not displayed in the browser in the error overlay:

As a counter-example, the CRA errors in the overlay displayed locally:

I didn't find the original PR introducing this in CRA, but I found this PR discussing the feature: https://github.com/facebook/create-react-app/pull/7091

Edit: Maybe this is the original PR? https://github.com/facebook/create-react-app/pull/5903

This code from the PR looks like it could plausibly be the async handling of TypeScript errors (with modifying the webpack compilation stats and sending the errors and warnings to the webpack socket):

const messages = await tsMessagesPromise;
clearTimeout(delayedMsg);
statsData.errors.push(...messages.errors);
statsData.warnings.push(...messages.warnings);

// Push errors and warnings into compilation result
// to show them after page refresh triggered by user.
stats.compilation.errors.push(...messages.errors);
stats.compilation.warnings.push(...messages.warnings);

if (messages.errors.length > 0) {
  devSocket.errors(messages.errors);
} else if (messages.warnings.length > 0) {
  devSocket.warnings(messages.warnings);
}

How has this issue affected you? What are you trying to accomplish?

If possible, I would like to see TypeScript error messages in the error overlay in the browser (like in normal CRA locally).

Link to sandbox: link

Your Environment

Ref (maybe related?): https://github.com/codesandbox/codesandbox-client/issues/1303

Reproduce the missing overlay errors with the linked create-react-app v3 sandbox, then compare the behavior with the referenced Create React App PRs and the related issue. Done means TypeScript errors from the sandbox appear in the browser error overlay, like the locally displayed CRA errors.