Update won't install. How to fix (or should I bother)?
How can I get the update installed?


It is possible that the device has been rooted or tampered with by your friend, such that it no longer passes the integrity check before OTA updates. Whether this is the case is trivial; if you want the update regardless, you will have to resort to flashing a factory image which requires you to unlock your bootloader and lose data on the tablet in the process.

However...


  Alternatively, is this even something to worry about?


Nexus 7, even the 2013-released version you have, is long obsolete as seen by Google, and therefore misses out on a large amount of security patches since its last official update in 2016. Updating it to a Feb. 2016 patch will not change how vulnerable it is by much.

To get an up-to-date security patch, unlock your bootloader, flash a custom recovery then a recent custom ROM such as LineageOS, which would bring you to both a newer Android version and security patch at the same time (Android 7.1.2 and patch set Oct. 2017 at time of writing).