The situation of TKD is very similar to that of Shotokan karate (and indeed since TKD comes mostly from Shotokan karate).
In Shotokan, you will occasionally see some weapons being taught, such as the nunchaku, tonfa, sai, and sword. But those weapons aren't in Shotokan's syllabus. Each instructor had to learn them from someone who knows Okinawan or Japanese kobudo (classical martial arts).
It's usually taught as supplemental material, not part of the official syllabus, depending on what your instructor happens to have gone out and learned on his/her own. Weapons could be part of the syllabus for that particular school only, but not all Shotokan schools.
And this is the same as what you'll find in TKD.
In my old TKD school, my instructor was adamant that weapons had no place in his school. To him, TKD was for defense purposes only, and weapons were offensive (in both senses of the word). At least to him.
I later learned that other instructors see it very differently. They might point out that in order to defend oneself well against someone armed with a weapon, you better be familiar with that weapon. So it behooves you to learn how to use weapons.
Also, they point out that learning to control a weapon will magnify any weaknesses you have in your stance, strength, coordination, speed, timing, etc. All of that can benefit your empty-hand abilities as well.
Personally, if you're trying to decide between two TKD schools, one with weapons and one without, I think the choice of schools should almost entirely depend on how good their actual TKD (empty hand) skills seem to be, rather than valuing weapons training particularly highly. But that's how I would value them. You may see it differently.
Also, I've not found very good weapons instruction in TKD schools, I'm afraid. In fact, I'd say most are absolutely atrocious at it. My recommendation is to look around. See who's teaching what. Go to their classes and see for yourself what it looks like. There is a lot of good weapon instruction in many different martial arts. Such as: Okinawan kobudo, kung-fu, iaido / kenjitsu / kendo, Bujinkan ninjutsu, classical jujitsu (many have weapons), Filipino martial arts, or Silat. Have a look around.