Yes, the different main menu screen was the easter egg. I'm sorry if you found it underwhelming.... haha....
The stay option gives you a slightly different ending but it's like a minute of extra content at best, if that. You're not missing anything significant.
My mindset while making this game was to use it as a test bed for features I can use for my next games so some things in the game might feel underutilised. I do apologise for that.
I'm glad you enjoyed the game anyway, I probably won't be adding anymore content to this game, but I am working on a new game currently (which Itch refuses to index) called At Her Mercy which will have a much longer story and a lot more variety of choices.