The  phrases that begin with "let us/him/them/etc. are from the Greek third-person imperatives. In English, all imperatives are second-person commands, addressed to the one you are speaking to. In ancient Greek, the form is not only used for commands but requests. It is also used in the third person. This is transformed in most Biblical translations by changing it to a second person phrase because with "let him/them" do whatever the verb is. This distorts the form from third person to the second person making the subject of the verb an object.. This strains the meaning. It is better translated simply as "he must" plus the verb. A longer explanation here.

There are around fifty of Jesus's verses where he uses this "let" phrase. In English, all commands are the second person, given to the person addressed. However, in Greek, the imperative form, used for both commands and requests, can be "given" to an object. This is usually translated in the Bible into a second-person command beginning with "let...." followed by the command verb with the subject as its object. When we see a "let," however, we cannot assume a third-person command because a very common verb also translated as "let" in the sense of "let her alone." Adding the "let" turns the third-person command, a strong statement of desire, into a weak second-person command for something to be allowed. However, often the third-person form of the verb is disregarded entirely. This is true, for example, in the Lord's Prayer, which begins with a series of third-person commands.

There are two better approaches to translation. One is to add a helping verb "may" to the beginning of a declarative sentence. "May that name of yours be made holy. May that kingdom of yours show up." However, this makes the verb look a little-bit like a subjunctive, something that could or should happen. Perhaps the best approach is to use the helping verb "must." "The name of yours must be made holy. That kingdom of yours must show up." There is a verb in Greek that is translated in the Bible as "must," but it actually means "it is needed," so correctly translated there is no confusion between the verb and the verb form.