Time heals all wounds?

Matthew 5:23-24 
"Therefore, if you are offering your gift at the altar and there remember that your brother has something against you, leave your gift there in front of the altar.  First go and be reconciled to your brother; then come and offer your gift."


People always say "time heals, things will work out".
Does it really?
The passage above is often preached with a tone of judgement against hypocrisy; what I read is urgency.  Drop everything and reconcile first!  Time alone doesn't heal, all it does is allow a broken relationship to fade into the past.  The mind forgets but the heart doesn't.  Whenever my heart brings up unresolved conflict from long ago, I wish that I had handled things better and sooner.