refactor: Remove more promise awaits (#28733)

apps/meteor/app/integrations/server/api/api.js

apps/meteor/app/lib/server/functions/insertMessage.js

apps/meteor/app/lib/server/functions/sendMessage.js

apps/meteor/app/livechat/server/lib/Analytics.js

apps/meteor/app/livechat/server/lib/analytics/departments.js

apps/meteor/app/livechat/server/startup.ts

...ivechat/server/statistics/LivechatAgentActivityMonitor.ts

apps/meteor/app/push/server/push.js

...teor/ee/app/livechat-enterprise/server/api/departments.ts

...pp/livechat-enterprise/server/hooks/afterInquiryQueued.ts

...e/app/livechat-enterprise/server/hooks/autoCloseQueued.ts

apps/meteor/ee/app/livechat-enterprise/server/lib/Helper.js

.../livechat-enterprise/server/lib/QueueInactivityMonitor.ts

apps/meteor/ee/app/livechat-enterprise/server/startup.ts

apps/meteor/ee/server/apps/appRequestsCron.ts

apps/meteor/tests/end-to-end/api/livechat/06-integrations.ts