considering the fact that kurt admitted multiple times that most of his lyrics are random and meaningless and that he made them up on the spot

I'm not a Kurt expert, but I'm not sure he is telling the truth.

For Bleach and pre-Nevermind many of the lyrics were made up right before recording.

Then he started keeping extensive lyric/sketch/diaries in spiral bound notebooks. It's my understanding that's where much of the lyrics for 'Nevermind' and 'In Utero' came from. I don't have it but you can buy a copy ('Journals' by Kurt Cobain) on Amazon for $17.

This isn't Kurt's process, but it works for me - start a lyric notebook. Force yourself to write 12 lines of lyrics a day. It's OK if it's crap. Can't think of something? Try to pick the worst / dumbest line to start with... and see what you can answer it with. THEN before stopping for the day pick up a guitar and record a quick demo of the lyrics into a audio note app on your phone -- sing it twice in two different genres in the same take -- i.e. do rock ballad and then speed metal, or indie rock and melodic rap, etc. Let it sit for a day, and then go back and listen to it, and see which words/rhythms worked. Turn the volume down so you can't hear it well and see if you mishear any of your phrases as something better. Then repeat (writing 12 lines, recording, and then listening to them the next day). After a week or two you will probably have a few that you are interested in finishing / developing further

edit: bonus link - rehearsing/writing for Nevermind in a barn - https://www.youtube.com/watch?v=d3zAvvSl9QA . They were practicing hours a day for ~5 days a week. They kept forgetting songs/jams so they recorded/boomboxed some and videoed this.