I just... don't think it needs to be limited like that. Whenever you level, you can swap spells around, especially cantrips. I don't see a power problem with allowing a character who learned Fire Bolt as an Artificer from using the War Magic ability, especially since we can see that the "wizard cantrip" limitation does not exist for Valor Bard, nor did it exist on the Bladesinger wizard.
That the game text contains examples of both more and less permissive text buttresses my arguments
Level 6: Extra Attack (Valor)
You can attack twice instead of once whenever you take the Attack action on your turn. In addition, you can cast one of your cantrips that has a casting time of an action in place of one of those attacks.
Level 14: Battle Magic (Valor)
After you cast a spell that has a casting time of an action, you can make one attack with a weapon as a Bonus Action.
Level 7: War Magic (EK)
When you take the Attack action on your turn, you can replace one of the attacks with a casting of one of your Wizard cantrips that has a casting time of an action.
Level 18: Improved War Magic (EK)
When you take the Attack action on your turn, you can replace two of the attacks with a casting of one of your level 1 or level 2 Wizard spells that has a casting time of an action.
EK is distinctly limited to Wizard spells, while VB has greater freedom. One justification is that at level 10 Bards get access to cleric, druid and wizard spell lists: that's in the class core and doesn't rely on species choices, feats, or multiclassing.
It is a valid limitation if you want to add it, but I don't see a reason to add it, other than a strict RAW reading of the class feature.
(Emphasis mine.) My focus is of course to show which reading is best motivated by the words themselves. You touch on other motivations - power, equity, agency - which are valid. I see my arguments as helpful toward those goals, and the difference between us might then come down to approaches. Having found a consistent comprehension of the RAW, I can House Rule over it to produce more robust play. For example, I could House Rule that
a casting of one of your Wizard cantrips that has a casting time of an action
is replaced at our table by
a casting of one of your cantrips that has a casting time of an action
Although I wouldn't because I don't share your concerns in that regard, and I don't agree that EK needs that sort of equity with Bard. However, I will House Rule that EK War Bond contains the text
In addition, you can use a weapon you have bonded to yourself as a Spellcasting Focus to cast your Wizard spells.
And based on
@Sorcerers Apprentice's effective arguments up thread, I'll read that as covering their M and S components.