Italian meringue deflated after adding sugar
I've only ever made French meringue prior to this, and I've only ever used whole, separated eggs.
In this case, I used egg whites out of a carton (probably should note they were only a few weeks shy of their expiration date). I know fresher whites whip up better..

Older egg whites being thinner volume up better, but the drawback is being less stable (not so suitable for french meringue).  Polar opposites to fresh eggs.  However you are going to stabilise them by pouring hot syrup and keep whisking till it's cool, so no worries.
Note: carton eggs are usually pasteurised, so there is a slight performance decrease, but not enough to prevent from making a meringue

..but I definitely got these to soft/slightly stiff peaks when I whipped the whites on their own. However, when I added the sugar syrup the meringue turned liquidy pretty quickly. I kept it whipping for a bit but the texture remained pretty liquidy (I did use some cream of tartar to help stabilize, but to no avail, apparently).

Presume the cream of tartar was added when the whites were foamy.  It is mainly used as a stabiliser to prevent overbeating.
There are two camps on Italian meringue methodology:

Adding the syrup at soft peaks,
This is with the consideration of using a stand mixer or a suitably powerful beater at high speed to beat the whites to stiff peaks before it is stabilised, and continue beating until it has cooled.
Adding the syrup at firm/stiff peaks
This allows you to also whip by hand, hand beater or stand mixer, until cool.


I'm wondering which, if any, of the following is the likely culprit:
is there any general detriment to using the whites from the carton?
was it simply the lack of freshness? (I would be surprised if it was either just because the whites did whip up pretty nicely before I added the sugar)
could the syrup have been too hot or in some other way caused the whites to break down?
could I have overwhipped the whites before adding the sugar? the peaks didn't seem to be too stiff before that point, but I suppose it's possible
something else?

It's not clear in the OP whether:

The whites were beaten by hand, beater or stand mixer;
The temperature of the sugar syrup reached 'soft ball stage' (112°C - 116 °C or 234°F - 241 °F)
What speed were the whites whipped while the syrup was being added

One or more of these issues may have caused the collapse.