I get the basic argument: if 1 of 4 voices was in parallel fifths, and the only objection to parallel fifths is their lack of part writing independence, then such a texture should be simply regarded as 3 independent voices, and if 3 independent parts is OK, the 3 independent voices with a 4 voice in parallel fifths should be OK too.
Here is the problem: the fairly common point...
Parallel octaves and fifths are not allowed in four-part writing because it undermines the independence of the two voices.
...is more of an apology for the strict part writing rules of the common practice era.
The fact is parallel fifths are prohibited, because the sound was undesired during the common practice era.
It isn't really an issue of part independence. If it were, then parallel thirds/sixths would not be so readily accepted.
The difference between parallel fifths and parallel thirds is relatively small. In a passage of parallel thirds, like this...
...it would be silly to say the two violin parts are truly independent lines. It's a conspicuous instrumental doubling more akin to an orchestration effect.
Aside for the purely aural difference between parallel fifths and thirds - some people like organum, some like Fauxbourdon - there is one technical difference between parallel fifths and parallel thirds. When playing parallel thirds the intervals qualities change between major and minor, are always consonant, and don't depart from the key signature. But when playing parallel fifth at some point you will either hit a dissonant diminished fifth or in attempting to make it a consonant perfect fifth you will deviate from the key signature.
So, those are two reason, which are not specifically about part independence, why parallel fifths are avoided: harmonic taste and maintaining consonances within a key signature.