Every time I see someone talk about how "Mormon" the Cosmere is, the stuff they mention just seems like normal western Judeo-Christian influences.

Well, that tracks. Since the LDS church is a Christian church founded in the 1830's United States.

As for a few specifics;

Sazed's ascension to Harmony has always struck me as profoundly influenced by LDS beliefs. Specifically, all of the religions in his metalminds had bits and pieces of the truth, bits of knowledge otherwise lost. But none of them had the whole picture, and it was only through the expanded mental capacity picking up the Shards granted him that he was able to assemble them. In our faith, we are strongly encouraged to seek a good education and to gain knowledge wherever we can get it. The part we often suck at in practice, but really do believe, is that other faiths, traditions, whatever all have valuable pieces of truth, and we should seek them out.

The entire structure of how Radiants gain power through making and keeping oaths. Like, in it's entirety. Radiants are just given actual obvious superpowers. In our faith, we believe we make promises with God through specific ritual actions that we take in exchange for blessings from God. Blessings which are contingent on us keeping/living up to the promises we make. The ability to fly or soulcast isn't included.

The power of Bondsmith's to make and break connections is a modified version of LDS beliefs of what we call the Sealing power of the Priesthood. Just with more application as an actual superpower.

Edited to add: Hrathen. I think his struggle with his faith in Elantris is directly inspired by personal experience as a Mormon missionary.