If the HD fails in the future, will it be likely due to the sectors in those first 300 GB partition?
No. Failure will be more likely to be detected in that portion of the physical drive, but only because that is the portion of the drive that you are using. Failure can still happen in the unused portion, but since it isn't being used, you will never know unless you instruct the drive to run a full-drive surface scan and look at the results of that scan.
SSDs have a variant of this failure mode, however, because specific physical flash locations can only take a certain number of writes (with modern drives this amounts to a few thousand writes per physical flash location) before the flash memory wears out and can no longer accept further writes to that location. For this reason, SSDs employ (among other things) a technique called "wear leveling" that effectively moves data around to spread the writes across different physical locations. Wear leveling allows SSDs to have a significantly longer lifespan than they otherwise would, to the point that, in spite of what you might read elsewhere, flash failure is not a major concern with modern SSDs. This does not apply to rotational HDDs.
If in fact the life of the HD can, on average, be extended if the usage is smoothed out across the entire HD, how should I go about doing that? Will extending the partition to fill the entire 1 TB help?
No; even if HDDs benefited from some form of load smoothing, merely expanding the partition would not accomplish that if you aren't using the additional storage space. Since you would still be storing the same data on the drive, nothing would change except a single value (a few bytes) that tells the operating system how far the specific partition extends.
Also, with the data encoding used by modern hard disks, it doesn't really matter what is being stored in a particular location, because everything gets transformed to a jumble of (and I am simplifying here) "ones" and "zeroes" when written to the physical platter that actually holds the data. As an example, a large section of a file containing only zeroes is not written as a blank or all "zeroes" to the physical drive platter, because writing it like that would make it very much more difficult to read back reliably. The same goes for space that you aren't using at all and never have; it will appear to software as being (likely) all zeroes, but physically it is something very different.
TL;DR:
There is no reason whatsoever to be concerned because you aren't using the full capacity of the hard disk drive.
Whatever failure might happen would happen the same way regardless of whether you store only a few megabytes on the drive, or are using the whole drive.