Do not worry about emulated block size, because ssds only use that to report to the filesystem, no sectors exist on solid state media.
Well SSDs have flash pages, which are essentially sectors as they are the smallest unit you can read/write, and they're usually 4096 bytes on NAND flash. My understanding is to write to a flash page you have to erase the whole block of 32+ pages before writing, so if a 512e SSD doesn't receive the rest of the page in time (e.g. with IO scheduling, 512 byte fs sectors, unaligned partitions, or bad luck), then the controller has to do a read-erase-program across the whole block of pages (although wear leveling will probably read + copy the modified page to elsewhere instead), instead of just a single program operation on one page (assuming a properly trimmed SSD).
So IMO, 4Kn is more important on SSDs, yet every vendor sets 512e by default. Luckily quite a few NVMe SSDs (not Samsung), let you change the logical block size with an nvme format command.
But more related to the OP, HDDs are a lot slower and don't have to erase multiple sectors before a write, so it's not as important, however partition alignment is still important but usually handled automatically in every modern partitioning tool. Sometimes you can switch a HDD to 4Kn with hdparm, but this isn't common, and you've said your HDD is dead anyway, so it won't help.
Also, sorry about bringing you into this, I have strong opinions against 512e SSDs haha.
