The Biur Halachah to Siman 132, outlines the order of preference in saying Kaddish, and he adds explicitly that precedence for serving as Shaliach
Tzibbur has the same rules as that for Kaddish.
Avel during shiva or sheloshim.
Then, Yahrzeit for a parent takes priority over a regular avel.
The FIRST YAHRZEIT trumps an avel in sheloshim.
Kaddish for other relatives is no obligation whatsoever, and he therefore has no claim to be the shaliach tzibbur.
Some shuls reverse the order of priorities of sh’loshim vs. yahrzeit, because of the reasoning that the sh’loshim has thirty days, while the yahrzeit has only one.
As for the person who comes to shul just for yahrzeit vs. the avel who
is a regular, the regular even if a lesser chiyuv has precedence.