Can I increase the reliability of my hard drive by using less of its' capacity?

Your assumptions are wrong.

The drive has a fixed physical format that is made up of physical sectors, in your case totalling 3TB. By formatting the drive you are effectively clearing the data in those sectors but you are not rearranging them or changing their physical size or layout in any way whatsoever. By formatting at a lower capacity you would simply be wasting the extra space, it would exist but would simply not be being used for anything at all.

Formatting does not change the physical sector size nor space between tracks.

As to unallocated sectors, I believe that drives have a preset number of reserved sectors for reallocation and the drive has no way of telling whether sectors are used by the operating system. SSDs do have a feature called "trim" that can tell the drive the sectors are clear to be wiped but this is something slightly different, spinning platters lack any kind of similar feature. Un-formatted sectors would not be being used for bad sector reallocation.

By formatting a 3TB drive as 1TB you are simply preventing yourself from using 2TB of space and thus wasting 2/3rds of what you spent on the drive. Your 1TB partition would be sitting in front of 2TB of empty space.

The error correcting algorithms used by hard drives, like all other error correcting algorithms, do have a chance of failure. There is an extremely remote but existent risk that the hard drive may have written different data than you sent it. With capacities getting into the trillions of bytes, this chance statistically increases.

But you don't buy yourself extra protection from this by leaving space unused. You do obtain extra protection by storing redundant data, either by using a PAR/PAR2 tool or writing the data multiple times.

However, if you are worried about this, you might as well put the effort into copying the data into another physical drive which has additional benefits like protection of mechanical failure and such.

Not using space would give you extra chances in the event of the hard drive head scraping the surface due to it being struck. But think about it - unless you have a method to space your data evenly sparsely around the disk, assuming you are using this data, the head is likely going to be on or near tracks that contain your data. The days where the hard drive controllers would respond to a "park" command to "get out of the way" of data are long past.