How can we reconcile the desirability of praising Allah and sending blessings upon the the Prophet before supplication for it to be answered, with the fact that the Prophet (peace be upon him) did not begin some of his supplications and remembrance (dhikr) with that?

Scholars are in agreement regarding the desirability of beginning a supplication with praise of Allah and invoking blessings upon His Prophet (peace be upon him), and concluding it with the same. This is considered among the means of its acceptance, but it is not a condition for the supplication to be accepted. Therefore, whoever does not invoke blessings upon the Prophet (peace be upon him) in their supplication is neither sinful nor will their supplication be rejected, especially if it is sincere and free from impediments. The supplications mentioned in the Prophetic traditions that do not begin with invoking blessings upon the Prophet (peace be upon him) indicate the permissibility of this.