[ Regarding the "pushing" I am not sure, as the aircraft is on the correct vertical profile (vertical deviation centred) but the GA altitude is (let's say) above Your current altitude so technically nothing should happen
Correct. Descent mode cannot engage unless FCU altitude is below the present altitude. In case after GS capture the OPCLB mode gets engaged then push VS and follow capturing GS from above procedure by setting a suitable VS like -1800.