What is the secret behind Allah Almighty’s saying: (Except for those who show patience and do righteous deeds), and why did He not say: "Except for those who believe"?

The verse "those who patiently endured" refers to the believers, because patience is one of the requisites of faith. They were described with patience rather than faith to contrast their state with that of the disbelievers, who were mentioned as "despairing, ungrateful." Thus, it was fitting to describe the believers with patience, awaiting relief, in contrast to the despair and despondency of the disbelievers.