I see it more as a blessing. Similar to the nerves in your fingers (nociceptors) that alert you that the stove is hot, and so you remove your finger.
If it weren't for those pain receptors you could easily harm yourself quite critically. This is a disease called Congenital Insensitivity to Pain (CIP).
Sensitivity to sin is similar in that it alerts you to harm, and that harm burdens your heart.
As the Prophet (saws) said:
"wrongdoing is that which wavers in your soul"
"wrongdoing is that which wavers in the soul and causes uneasiness in the breast".
Where a sign of CIP is a lack of sensitivity to physical pain, a sign of a diseased heart is a lack sensitivity upon committing sin.
Both absences of sensitivity require the invocation of a doctor, and the doctors of the heart are the scholars of Islam.