Ok, let me rephrase. Sure, it's an Arcane Firearm, as written, but it's not a weapon per se.
Besides, it does say it. It turns a wand, staff, or rod into a spellcasting focus for your artificer spells and that it's called an Arcane Firearm. In a manner of speaking it does say it's a firearm, but it doesn't say if it uses the statistics of a pistol, a musket, or some other variation of firearms or whether this Arcane Firearm has its own statistics in a similar manner to other firearms. It doesn't list what kind of ammunition it uses, or whether it has some sort of loading rules. An arcane firearm doesn't exist elsewhere in the rules.
If an Arcane Firearm was a weapon it would have statistics, such as a damage die and type (B/P/S or some energy type maybe) related to its type, a range, and other properties associated with actual weapons. You cannot make an attack with the Arcane Firearm without casting a spell, because the rules don't tell you how (I guess you could use it as an Improvised Weapon, in melee, but that's not the same thing). Do you see my point now?
Basically most of the text is just "empty words", or flavor. As you put it, the feature is rather weak. But it's flavorful, and that's not always a bad thing.
Combined with TCE Bladesinger (I checked; it works) I could use Bladesong, make an attack with a Rapier and, starting from Bladesinger 6th level, cast one of my Cantrips (it doesn't specify from which class it has to come from), and use the Eldritch Cannon with a bonus action. Since I'd need to hold the Arcane Firearm in the other hand, I couldn't wield the Eldritch Cannon in hand, but that's not a problem. It could just be what it's good at: a "Gadget"
Cantrips are still spells (in fact, they are "0th level spells"), so they'd get the +1d8 extra damage as well.
So, let's assume I was an Artillerist 5/(TCE) Bladesinger 6, and I had a Rapier, I could make the following attack routine every turn:
Attack Action: 2 attacks. One attack with a Rapier (1d8+Dex+maybe something extra), cast one cantrip in place of another attack (let's say Fire Bolt; at 11th level, it would deal 3d10 + the extra 1d8 fire damage from Arcane Firearm), and then a Bonus Action to activate Eldritch Cannon (as a Flamethrower, a Force Ballista, or a Protector).