But I would say that "presence" much better encapsulates what the WT teaches about Jesus' parousia, that is, an invisible presence that only a few know about,
Hmm...I am surprise to here you say that. Maybe I am misunderstanding you but are you saying his 'presence' and 'coming' are two different events?
That is not how I understand it and that is they may be two different words being used but they are referring to the same event/time.
Starting with this verse that uses the word 'presence', this is the way I view it:
[However, brothers, respecting the presence of our Lord Jesus Christ and our being gathered together to him,]
There it connects their being gathered together to him at the time of his 'presence'. Matt 24:31 shows when it is that they are gathered together to him, which is at the time of his 'coming' as shown in verse 30.
[30 And then the sign of the Son of man will appear in heaven, and then all the tribes of the earth will beat themselves in lamentation, and they will see the Son of man coming on the clouds of heaven with power and great glory. 31 And he will send forth his angels with a great trumpet sound, and they will gather his chosen ones together from the four winds, from one extremity of the heavens to their other extremity.]
Likewise:
[15 For this is
what we tell YOU by Jehovah’s word, that we the living who survive to the presence of the
Lord shall in no way precede those who have fallen asleep in death; 16 because the Lord himself will descend from heaven with a
commanding call, with an archangel’s voice and with God’s trumpet, and those
who are dead in union with Christ will rise first. 17 Afterward we the
living who are surviving will, together with them, be caught away in clouds
to meet the Lord in the air; and thus we shall always be with the Lord.]