Does Story Mode even save completion times?

It sadly doesn't, but you can still take a screenshot of the final time as partial proof, or even record the whole thing if you can.

 the site focused only in times that got saved to memory.

any mode that has stuff that doesn't get saved was never considered because it's harder to keep track of.

I have kept a full set of story times.  We currently woudldn't be able to add an actual ladder for it here because it would require database work.

I could always throw something together on my website.  We could also simply post times in this thread to compare them.

That'd be a good compensation. I know I've done a few Google Spreadsheets myself, so probably you can do the same ?