It is a funny question – if one wants to recite Desika stotras, how can that person be without any association with Ramanuja tradition? Adiyen is unable to understand how one who does not belong to Srivaishnava tradition will have the desire to recite Desika stotram. If one learns Paduka Sahasram and desires to recite, it shows their affinity for Desikan. This could mean they are also interested in Ramanuja sampradaya.
There are several stipulations for reciting at temples and one should follow the relevant stipulations to recite at the respective temples. In the temples at Divya Desam, other than authorized goshti, no one is permitted to recite. But in the newly constructed temples, anyone can recite any shloka.