Yeah, I was going to make a long-winded reply to the OP, but you captured the essence of what I was going to say in a couple lines.
There's really no "balance" issue with the ACKS Mage. They are not supposed to be the same as a Fighter, and a Fighter is not the measure of what other classes should be.
Take, for example, the Cleric; you laud the class for having the benefits of the Fighter (e.g., heavy armor, weapons), but the Cleric is, at best, a poor imitation of a Fighter at 1st level. Fewer hp, worse weapons, likely lower Strength as Wisdom is their Prime Requisite, no spellcasting, much poorer weapon selection...they're just like a Fighter, only worse!
What about the Priest(ess)? I'm playng one in a PbP campaign right now. All the downsides of a Mage, but none of the spectacular spells (e.g., Sleep, Charm Person, Summon Berserkers, etc.)! My best spell has been Protection from Evil. Tremble before me!
The underlying difficulty with these kinds of analyses is the presumption that combat is the measure of a class. But most classes trade away combat ability for the capacity to do other interesting and fun things outside of combat. Pretty much all classes can still contribute in battle, but it's not their focus. In the case of the Mage, extra Proficiencies and Languages from high Intelligence help carry the class. If you're looking for extra magical "oomph!" I'd suggest looking at Proficiencies like Black Lore, Familiar, Magical Music, Mystic Aura, and Prestidigitation. All will add "magical" ability without the addition of a new type of unlimited spellcasting. If you're not as fixated on being magical, Proficiencies like Healing, Loremastery, Magical Engineering, and Soothsaying also provide plenty for a Mage to do after they've cast their limited spells.
Back to your actual proposed Cantrips, I say no, I don't think they're game-breaking. I think unlimited 0th level spells breaks verisimilitude and potentially harms ACKS underlying economic assumptions, but I don't think they immediately make the class more powerful than all others. Personally, if I went with them, I would avoid damage Cantrips. 1st level spells can't do huge amonts of damage, and can only be cast once per day to start. Unlimited 0th level spells, even at 1d3 damage, would very quickly overshadow spells like Magic Missile. If you're worried about some kind of balance, it's probably better to restrict Mages to attacking with physical weapons just like everyone else, and leave 0th level spells for other types of effects.
EDIT: ...aaaand my reply was long-winded anyway,