Not necessary.
The USAF has a few hundred aircraft in theater...Qatar, UAE etc.
They can provide cover in transit.
A CVN would NOT be running flight ops while transiting this narrow channel.
I really have no idea other than my own flightdeck experience, as dated as it is. Seems there can be only two possibilities. Either they launched prior to entry and recovered on the other side, or launched and the squadrons landed at a nearby base until she passed the small channel. Either way, you're quite right, along with the normal escort, there's plenty of air-cover available.