Let us make clear that the problem

>If proton spin emergence from quarks and gluons is mysterious, why is silver atom spin not?

is a modelling problem. The spin both of the proton and the silver atom is measured and known to identify them.

John's answer covers it, the  energy carried by the virtual quarks and gluons within the proton are much larger than the energies carried by  the virtual protons neutrons within the silver nucleus , and more so of the virtual electrons and the photons they exchange between them and with the nucleus . This means that a central-potential-well  approximation will work to first order for nucleons and atoms,  and that makes modeling both the nucleus and the silver atom successful, i.e. fit the observations. This is false for the interactions within a proton, no central potential well can be devised to model it.

Modelling QCD interactions cannot be reduced to such a simplified model, and "mysterious" just reflects the difficulty of the mathematical tools necessary and the observations necessary to explain how within a QCD model the spin arises. [Here is a talk][1] of some time ago but it sets up the problems. It is a current matter of research both theoretically and experimentally, as can be seen [here.][2]  .


  [1]: https://www.bnl.gov/riken/rpcsm/linked_files/talks/pdf/Boer.pdf
  [2]: http://physicsworld.com/cws/article/news/2014/jul/11/gluons-get-in-on-proton-spin