No, cells have active DNA repair mechanisms working all the time. They also have transcription ECC that attempts to detect and/or correct transcription errors.

No, cells have active DNA repair mechanisms working all the time. They also have transcription ECC that attempts to detect and/or correct transcription errors. Some critical protein coding sequences are arranged to make some errors harmless, or are duplicated, etc. Selection pressure plays a large part here. Waterbears (tardigrades) have exceptional DNA repair mechanisms — far superior to ours in some ways — because they must survive constant dehydration and rehydration cycles. Without the ability to repair what we would consider catastrophic DNA damage they'd have gone extinct long ago.