Flavor can be changed without altering the mechanics
There are no (sub)classes that have to cast through items, but how you flavor your spellcasting is up to you.
Spellcasting works by expending spell slots that return after either a short rest or a long rest depending on the class in question. It is easy enough to visualize your available spell slots as bracelets. Mechanically you would still have spell slots but they are represented as bracelets for flavor. They can magically replenish after a rest or you could roleplay crafting them, for prepared spellcasters who prepare their list of available spells each morning this works great too.
Alternatively, or additionally, spellcasting classes have the option to use magical focuses or component pouches to substitute a spell's material component. What form these items take doesn't really matter so they could be bracelets.
You did mention something that would not really work using this approach:
The bracelets would be destroyed after use, the power and type of the spells would differ depending on the material and technique used to make them
This suggests you would have to prepare each individual spell beforehand, with the spell being used up unless prepared multiple times. Spellcasting in 5th edition does not exactly function this way, but unless you really want this mechanic it is easy enough to say you have the appropriate bracelet for a prepared spell when you want to cast it instead of having to plan ahead for each spell you think you will cast that day (which would be very limiting compared to other spellcasters). You can still describe each bracelet as you cast the spell, giving you all the flavor you seem to be looking for, but you won't get the exact mechanics as you described.
If you do want the exact mechanics, you would indeed have to homebrew an entire class. You could start by looking into the Vancian Magic system that was used in earlier editions of DnD, where each spell had to be prepared individually and would be used up once cast unless prepared again. Keep in mind this system is much less flexible so it's advised to compensate for this.