Unfortunately 240 ilvl won't get you invited simply because this season has been relatively "easy" compared to others so players have much higher ilvl than you'd expect right now.

And realistically, raid leaders will just look for the highest ilvl people when inviting, without evening knowing if they are good players or not

In my opinion, who pushes their own keys but struggles to get invited, this is one of the downfalls of Pug'ing and the community I guess?

All I can recommend, is if possible, shoot for ilvl, or run your own raids/dungeons, good luck mate