Aluminium Aerobars on Carbon Handelbars
I'm not a composites engineer. However, I'll attempt to explain.
Some carbon bars can take clip-ons
If a carbon fiber handlebar or a carbon fork steerer can withstand the force of your stem clamping on it, then chances are a handlebar could also withstand the force of aero bars clamping on it, assuming it was designed to do so. I've found several carbon handlebars that specifically claim to be compatible with clip on aero bars in general, like the Bontrager XXX aerodynamic handlebar. This bar's tops are flattened for aerodynamics (laugh if you want, but it actually makes a small but measurable difference). I suspect the bar would take clip ons in the rounded center section. The ENVE SES aerodynamic handlebar is compatible with ENVE's own clip ons, which are designed to clamp onto the flat section. Specialized makes its own version for International Triathlon Union (ITU, where drafting is legal) riders of the Venge aero road bike, which has aerodynamically shaped bar tops as well. And, naturally, many carbon bars are round, and some of them, like this Bontrager Pro handlebar, claim compatibility with clip on aero bars.
As a counter-example, consider the 3T Superleggera (Italian for super-light) handlebar. The Limited and Team versus weigh under 190g per bar and explicitly say they are not compatible with clip-ons. The 240-250g Pro version explicitly says it is compatible with clip-ons.
If they don't say they can, assume they can't
I'd wager that if a carbon handlebar's literature does not specifically state compatibility with clip on aero bars, it is not compatible. I would assume that its clamp area is only reinforced right where the stem clamps. So, the presenter in the Global Triathlon Network video isn't entirely correct, but she's definitely not wrong.
Recall that carbon frame tubes are not designed to withstand clamping, and frames can be damaged if just clamped in a repair stand. If a carbon handlebar wasn't designed to safely take clip-ons, it might not have the necessary wall thickness or other structure to not crack under the clamping force, and I would assume that it was not designed to withstand clip-ons.
Why would not all carbon handlebars be able to take clip on aero bars?
@mattnz linked to The Rules, which is basically a tongue in cheek style guide for performance-oriented road cyclists. He correctly stated that Rule 54 says not to mount aero bars (unless you're in a mountain time trial). The Rules are definitely intended to be tongue in cheek. However, aside from fashion and peer pressure, there's a substantive reason for roadies not to mount clip on aero bars: your control over your bike is reduced considerably when in the aero bars. Hence, I'd argue that you should not use aero bars if in a paceline or even if you are riding with a large group of riders, barring explicit prior agreement (e.g. you are doing team time trial practice and you all agree that the first X riders will stay in aero bars). If you are in a sanctioned mass-start road race, you're not allowed to have clip-on aero bars at all.
I'm not sure how high the demand for clip-on compatibility is among many drop bar riders. In my experience, most riders don't mount them. Hence, some manufacturers might assume they don't have to design their carbon bars to accept aero bars. This might not be a sound assumption.
What about light aluminum bars?
This discussion raised a question for me: should we assume that aluminum handlebars are safe for clip-ons? I recall that the lightweight aluminum handlebars of the early 2000s may have had specified weights of 220g or less, whereas claimed weights seem considerably more for most aluminum bars today (e.g. the lightest alloy Bontrager handlebar seems to have 260g claimed weight).
I do not recall if lighter aluminum handlebars were rated for clip-ons or not. It does seem possible that the lighter bars may have been pushing the limits. If they were aggressively butted (i.e. their wall thickness was different through the length of the bar), then it's possible the area by the stem clamp got butted to a lower thickness and that it wasn't safe for clip-ons. However, I don't really have a way to confirm this right now.
In any case, I suspect that people looking to buy a light bike today would either use carbon handlebars, or else they might not bother with light handlebars entirely. There may no longer be a demand for or a supply of ultralight aluminum bars. The ones that remain probably do have sufficient wall thickness to accept clip-ons.
Side note: internal view of a carbon handlebar
I suspect the stem clamp area on carbon handlebars is much thicker than the rest of the bar. I'm pretty sure I see that in this video by Luescher Teknik, an Australia-based carbon fiber testing and repair expert who cut his teeth in the aerospace industry. (Vid should start at about 6:10 in, and it shows a cut-up 3T carbon handlebar.)