What is the ruling on اطلاق (applying) the word "mushrik" (polytheist) to the perpetrator of shirk (polytheism) while refraining from ruling him a kafir (disbeliever) until the establishment of the hujjah (proof), and do the sayings of the imams of the Najdi Da'wah support this?

The terms "shirk (definition)" (polytheism/associating partners with God) and "kufr" (disbelief) are equivalent in their linguistic application and rulings. Declaring someone a disbeliever (takfir) is a legal ruling that belongs to Allah and His Messenger. A specific person cannot be declared a disbeliever unless the conditions are met and the impediments are absent. A Muslim may commit an act of disbelief, but may not be declared a disbeliever due to the presence of an impediment such as coercion or ignorance.

A specific Muslim is not declared a disbeliever merely because of an act he committed, until the proof (hujjah) is established against him. Not everyone who falls into disbelief is actually deemed a disbeliever. A statement may be an act of disbelief, and its utterer may be declared a disbeliever in general terms, but the specific person who uttered it is not declared a disbeliever until the proof is established against him.

It is not permissible to apply the term "kufr" or "shirk" to a specific individual except after the conditions are met and the impediments are absent. Declaring a specific person a disbeliever means that he is, in fact, a disbeliever.