In my opinion, all weapons should have 3 weapons attacks : 1 common to all weapons, 1 common to clearly defined subgroups, 1 common to other subgroups that may vary a bit more.
More advanced weapons (Martial) *should* have more advanced combat capabilities. Something along the lines of the below:
Simple weapons: +1
Martial weapons: +2
Two-handed weapons: +1
Off-hand weapon (any type)/Shield: +1 (different from main hand so as not to be penalized for using an item in the off-hand)
I see no good reason for not adding a similar action to shields.
As for balancing, using each once per rest seems overly artificial and nonsensical. Also, I assume, very exploitable (something that adds an inelegant cheesy feel of the game) either by dual-wielding different weapon types or by simply swapping weapon types (might be wrong in how it works in-game). This is homebrew, so why not just adopt "DOS-style" cooldowns. 1-2 turn(s) universal cooldown for all special actions after any special action is used. Restrict the use by requiring a bonus action. Thus we would achieve opening up an interesting system for wider use while also balancing/"uncheesing" the system.
