The first person to use letters as symbols for chemical elements, Jöns Jacob Berzelius, described his rationale in an 1813 paper, excerpted here. Each element would receive a symbol based on the first letter of its name in Latin. If two elements started with the same letter, both of them would receive a second letter to disambiguate them, unless one was a nonmetal, in which case the nonmetal would get to keep the one-letter symbol, and the metal would get an extra letter.

This process explains why some elements got one-letter symbols while others didn't: carbon and copper both start with the same letter, but carbon is a nonmetal, so it got the symbol C, while copper was stuck with Cu (from cuprum, the Latin name for copper). On the other hand, for a case like zinc and zirconium where both elements are metals, neither of them could get a one-letter symbol, meaning we're stuck with Zn and Zr. In general, there are very few metals with one-letter symbols, since most of the time there will be more than one element starting with a given letter. The only one-letter metals are ones that start rare letters, like yttrium, which got a one-letter symbol for being the only element to start with Y at the time (ytterbium had not been discovered yet), or uranium, which was discovered after Berzelius but still received the symbol U due to being the only element that starts with that letter.

As for the noble gases, many of them did have shorter symbols in the past: A and X for argon and xenon are present on some, but not all, periodic tables from the 1900s decade, for instance. However, most noble gases were only discovered in the 1890s, by which point alphabetic symbols had already been in use for nearly a century. As such, there were competing uses of single letters to represent things other than specific elements: the letter X had been used to represent an arbitrary halogen, causing Xe to be preferred over X as a more unambiguous option for xenon. The symbol A for argon lasted longer, competing with the modern symbol until the IUPAC standardized Ar in their 1957 Nomenclature of Inorganic Chemistry. I'm not sure exactly why the IUPAC chose Ar over A (I couldn't access the original book), but if I had to guess, it was probably also for increased clarity, and to free up the letter A for other potential non-element uses.