Objects do not trigger Opportunity Attacks
Your assessment is correct. The rules for OA state that (emphasis mine):
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
As you can see the rules are very clear. Unless something is a creature it can't provoke an OA.
Further, the turret created by the artillerist is not a creature but an object. From the UA:
The turret is a magical object that occupies its space and that has crablike legs
Thus, the turret also can't provoke an Opportunity Attack.
It may seem reasonable for a DM to want to change the Artificer turret to a creature (likely a construct is appropriate). However, do keep in mind that this changes a lot more than just Opportunity Attacks. There are a number of spells and abilities that can only affect either creatures or objects or work differently with them.
With the release of Eberron: Rising from the Last War nothing changes
This question was asked about a playtest UA version of the Artificer which has now been officially released in Eberron: Rising from the Last War. The wording of the artillerist turret has changed, it is called an eldritch cannon now. However, it is still an object:
The cannon is a magical object.
Therefore, this version of the artillerist cannon also does not provoke Opportunity Attacks, as explained previously in this answer.
By RAW Tiny is anything from bottle/lock size and downwards
Going by RAW, the smallest you can get for a 5e object is Tiny, even if it's smaller than a bottle or lock. A blade of grass or strand of hair has the same stats and does the same damage as a bottle or ball bearing.
There are currently no sources for 5e which support any classification for objects smaller than the examples listed in the PHB.
Before I get into the meat of the answer, it needs to be stated: You're the DM, you can do whatever you want. Especially when it comes to reskinning without changing anything mechanically. (Even players can do that.)
Notice that there's no limit on ammo mentioned, suggesting that the intent is not for the kobold to be carrying the large rock(s) around. While we don't have a clear statement here, it seems unlikely that the kobold magically produces large rocks at will. Therefore, it can be presumed that it's picking them up from its immediate surroundings. That being the case, it seems reasonable that one large heavy object is much like another, and the kobold can use whatever comes to hand.
I assume the attack says "rock" primarily because there will generally be rocks in the vicinity of most places you'd expect to see a kobold, and "Dropped Large Heavy Object" just doesn't roll off the tongue.
It should be noted that, while you as the DM are free to rule however you please, the "Dropped Large Heavy Object" attack should probably have the same stats as the "Dropped Rock" attack regardless of which object you actually use. It's already a fairly powerful attack for a CR 1/4 creature, so giving it a damage bonus because the kobold picked up a spiky rock (for example), might not be the best idea.