Will a secure erase note and log any bad sectors on a drive (rotational)?

Secure Erase has been removed from the most recent versions of MacOS, because it is not good for the SSD drives shipped in most Macbook Pro these days.

Years ago, the equivalent of secure erase, write Zeroes, was used for exactly what you describe. It also forced the drive controller to substitute a spare block (from its private stash of spare blocks) for any found to be bad.

These days, modern drive controllers "take names" of block that are reading bad, and when new data are supplied for a block going bad, a spare block can be substituted. so the drive controller already knows which blocks are going bad, all it needs are replacement data for that block number.

yes, it definitely is still the case today.

Drives have changed in waves, and the most recent wave was Hitachi inventing perpendicular magnetic regions on the platters. So the magnetic regions are now "standing up" instead of head-to-tail around a track.

That meant the MUCH more data could be crammed onto a platter, but the killer was, the data rate on a track went WAY up. That meant drive manufacturers needed to create new data separator chips. That meant they had to re-invent their ENTIRE product lines.

Every wishful-thinking pipe-dream kind of feature they had been thinking about went into the new generations of drives. Cheaper interface like Serial ATA? Sure, it's a new drive. Have the drive controller remember what blocks are generating read errors? Sure, it's a new drive.