A .tar archive is basically only the files cat’ed together, with a header and index added, right?
And a .tar.gz takes forever to modify, because it needs to first extract the .tar.
So why is there no archive format that just cat’es the compressed files together?


Archiving (catting files together) and compression are two different actions. This is true even in formats and zip or rar where the functions have always been a part of the same utility. They are separated in Unix because the Tape ARchiver wasn’t initially intended to produce a file on disk. The tar utility did eventually add compression but that’s not always desirable.
An archive of compressed files is just another archive regardless of the format. Compressed files is also a bit vague a term. Most video and music formats are compressed, so compressing them again doesn’t really add much value and can sometimes even produce larger sizes.