Y2K was definitely not only fear-mongering. Windows Systems did not use Unix timestamps, many embedded systems didn't either, COBOL didn't either. So your explanation isn't relevant to this problem specifically and these systems were absolutely affected by Y2K because they stored time differently. The reason we didn't have a catastrophic event was the preventative actions taken.

Nowadays you're right, there will be no Y10K problem mainly because storage is not an issue as it was in the 60s and 70s when the affected systems were designed. Back then every bit of storage was precious and therefore omitted when not necessary. Nowadays, there's no issue even for embedded systems to set aside 64 bit for timekeeping which moves the problem to 292277026596-12-04 15:30:08 UTC (with one second precision) and by then we just add another bit to double the length or are dead because the sun exploded.