A carrier isn't that complicated compared to a destroyer/frigate. There are a number of challenging areas, like running uptakes/access//stores/magazine lifts past an acceptable hangar arrangement, making sure the aviation ops spaces are in the right place and arranged properly, but in general there is space to do things. There are also some tunes to play in local strength and trade-offs between ship stability and aircraft operations and of course the power required to shift the beast can be a bit interesting to get into the water without shaking the aft end to bits.
A DD/FF on the other hand, has to meet most of the same challenges with far less space but the same variety of spaces to accommodate. Plus it will generally have multiple roles some of which tend to conflict with each other and require more/separate CS elements. A bit tricky to say the least.
CDP was essentially trying to play down the technical risk that some statements at the time were (wrongly) implying. The CVF risk is almost entirely programme (ie political) at this point.
I don't think the LPH/LPD took a forenoon to design either (in fact the LPD detailed design probably took longer than CVF!). Their problems are almost entirely ha'porth of tar induced and slowly being fixed (at a price of course).