fix: remove busy-wait loop and add missing await in EmbeddedChatApi

[AbhiramVSA]

Acceptance Criteria fulfillment

• Remove synchronous busy-wait (while (typingHandlerLock) {}) and unnecessary lock from
  handleTypingEvent — JS is single-threaded, no mutex needed
• Add await on fetch() in sendAttachment so the try/catch block catches network errors
  instead of silently bypassing them
• Add await on response.json() in getUserStatus, userInfo, and userData for consistent
  error handling matching the rest of the file

Fixes # (issue)

Video/Screenshots

Proof that missing await bypasses try/catch:

A minimal script (proof.js) using real fetch() against an unreachable host demonstrates the bug:

=== CURRENT CODE (no await — from EmbeddedChatApi.ts L1068) ===

ERROR ESCAPED past try/catch to caller: fetch failed

The try/catch inside sendAttachment NEVER ran.

=== FIXED CODE (with await added) ===

[INTERNAL try/catch] Caught: fetch failed
Result: error was handled internally

Without await, the try/catch in sendAttachment is completely bypassed — fetch errors become
unhandled rejections. With await, errors are caught as intended.

Busy-wait loop (before):
Line 366 contained while (typingHandlerLock) {} with a suppressed ESLint warning (// eslint-disable-next-line no-empty). This synchronous busy-wait is unnecessary in single-threaded
JavaScript and risks freezing the browser tab if the lock is ever stuck.

PR Test Details

Note: The PR will be ready for live testing at
https://rocketchat.github.io/EmbeddedChat/pulls/pr-1167 after approval. Contributors are
requested to replace 1167 with the actual PR number.

Replace Fixes # (issue) with the actual issue number once you've created it

#1166

[CLAassistant]

All committers have signed the CLA.

[AbhiramVSA]

This addresses #1166 — the changes are scoped to a single file
(packages/api/src/EmbeddedChatApi.ts). Happy to adjust anything based on review feedback.