Generally Royal names seem to fall out of favour post-1918 - having HMS Queen Mary vapourised at Jutland might have put a damper on the idea. Few WW2 ships were named after Royals.
I'll admit I am not really that well-versed in the History of the Royal Navy pre-WW2, and my knowledge of the Royal Family pretty much starts and ends with Queen Elizabeth II (rip). But my understanding is that, at least in the late 19th to the early 20th century, Royal Names were only used on battleships + battlecruisers. The exception being Ark Royal, which was named for an old 16th-century galleon rather than a direct royal.
Considering the lack of ships built during the interwar period due to the Washington Naval Treaty, wouldn't it be fair to attribute the scarcity of Royal names to the lack of battleships being built? The only ships that would've been eligible for a royal name were the KGV-class, of which 3 received Royal Names.
Anyway, not until the 'New Elizabethan Age' does it seem to have been 'cool' to use Royal names again.
If anything, it seems to me that using Royal names was used even less during QE2's reign. Before the new Queen Elizabeth-class CV, I can't think of any RN ships that used a royal name since WW2. The exception once again being Ark Royal, albeit I'd argue that they named it after the WW2-era CV rather than anything to do with the Royal Family.
I've recently read an Air Historical Branch paper that claimed that CVA-01 was to be named HMS Furious
All things considered, CVA-01 being named Queen Elizabeth makes sense considering it was the first capital ship to be built in QE2 reign. That would keep in line with the tradition (ignoring King George VI), except the names being now applied to CVs instead of Battleships/Battlecruisers. And considering her husband's naval roots, the Duke of Edinburgh makes sense to me for CVA-02. Considering this is backed up by Rebuilding the Royal Navy by David Brown and George Moore I'm heavily inclined to believe it.
I also decided to check the edit history, that blurb was added by a random IP on December 14, 2021. The source they cited is an archived website from 2007, hasn't worked since 2010, and was actually added in December 2007 as a source for the names of
QE and
DoE. That website only mentions the
QE &
DoE, not the
Prince of Wales or
Princess Royal at all. Add to that your point about there being no Princess Royal from 1965-1987, I think it's safe to say someone just made up the names. But since it's a pretty niche topic, it's flown under the radar and we've fallen victim to a citation cascade.
I honestly doubt CVA-03 or CVA-04 ever existed beyond the original 1960 Fleet Requirements Committee stating the Royal Navy needed 4 carriers to replace Victorious, Eagle, Ark Royal, and Hermes. Even before the 1966 Defense White Paper, I suspect the RN ditched the idea of 4 carriers and was only considering 2 carriers to just replace Eagle and Ark Royal.
Realistically- if CVA-03 or CVA-04 had been built, depending on when the (Audacious-class) HMS Ark Royal ( Possibly 1975?) was actually retired, I'd bet they'd reuse the name Ark Royal for a CVA just like they did for the Invincible-class carrier. For the other CVA, I'm pretty partial to HMS
Furious or HMS
Argus. But keeping with the royal theme tho, I'm partial to the name HMS Prince of Wales but feels a bit on the nose considering the new QE-class. Or even the HMS Duke of York. More as a tribute to the WW2 battleships at this point than for the affiliation with royalty, but still.
TLDR; People make shit up on the internet