I don't think it's like that. On YouTube they don't upload Pawandeep's every performance too because he is very popular and they want people to use the app too. Arunita's most of the performances are on YT.
About promos of this week I think they will upload before the episode. I have seen Arunita's post on Insta and it looks like she may sing Ye Galliyan ye Chaubara which was already sung by her and Arunita so this can also be a reason. I am not sure about it.
No it has nothing to do with wanting to eliminate her.
She sang Agar Tum Saath Ho, the rights are with T-Series. Sony only uploads those performance whose song rights it has. And T Series does not give any permission except channel rights.
You can check out older performances too...previous seasons. No T Series song has been uploaded by Sony TV. Even the Udit Narayan special, they did 3 songs from QSQT and spoke about the album too, but none of the songs were uploaded on YT.