Ever since the 3.0 update, or at least shortly after it, Steam seems to always believe there is a Steam client update to install. In game mode, it shows the "update available" icon even when there is no Bazzite update to install; if I click to install the update, it "installs" instantly and triggers a Steam client restart. Similarly, every time I switch to desktop mode, Steam pops up with an "installing update" window which also "finishes" almost instantly. Based on this behavior, I have to guess that there is no actual update to install and that Steam is bugged in thinking that there is.
This is not a huge issue as I can just safely ignore it and Steam still runs fine, but I figured you all might want to know about it anyway!