Our heartfelt sympathy goes out to

The sentence is OK. In daily conversation, the Britons and Americans I know would probably say nothing in reply to that statement, or maybe "Don't overdo it next time". An expression of formal sympathy is not required for such a minor mishap.

In general, conversational expressions of sympathy are not required for minor mishaps, except in the case of very small children. If the unfortunate event is intermediate in severity (getting fired from a job, death of a pet, being injured in a accident etc) people would probably say "I was sorry to hear about XXX". For something very severe (death of a relative or loved one) a friend or close colleague might say "I was so sorry to hear about XXX). Those formal (and formulaic) expressions where heartfelt sympathy goes out etc tend to be in written form and by more remote entities such as employers and would probably be regarded as insincere if spoken by a friend. You can even get ready printed "sympathy cards" with such wording. Many people, including me, find these deeply distasteful. Here is a particularly unpleasant example: