The auxiliary べき is inflected like a Classical -i adjective. Or I guess, more historically accurately, as a Classical -ki adjective.
(Note: This was introduced to me using the adnominal -ki form. Other materials might start from the terminal / dictionary form ending in -shi instead.)
|Irrealis / Incomplete
|Continuative / Adverbial
|Terminal / Predicative
(also "Dictionary Form")
|Realis / Conditional
- Note: Many Japanese grammars include a mizenkei for adjectives, as this is the canonical conjugation for verbs for constructing the negative. However, for adjectives, the negative simply comes after the adverbial, and moreover it is clearly separable as you can insert a particle between the adverbial ～く and the following ない.
Since べく is clearly the adverbial conjugation, yes, such phrases are syntactically and grammatically adverbial.
PS: Can anyone help file a bug report for how the furigana are not working when included within a table?