So... I have a problem. I knew that I won't be able to recruit new members if I get to the point when Delphine wants me to kill Paarthurnax. What I didn't know is that this starts w/o taking the quest, just at the start of the "Fallen".

So I ask. Is there a cheat or smth I can use to recruit those 3 new members, finish the quest and just say "Bye" to them or I'm stuck with this forever in my Journal...

If on PC, you can finish the http://www.uesp.net/wiki/Skyrim:Paarthurnax quest via console (not kill him, but it still counts as quest completed).
Then you can recruit the followers as normally. (I hope I didn't misunderstand the question)

Well I don't wanna "cheat" it, so both Blades and Greybeards will talk to me. I just want to finish the quest. It's design is horrible. I don't care if they don't wanna talk to me, I only want them to recruit 3 more members.

Then... kill Paarthurnax?

You are asking for a cheat however. The moment you stop playing the game in the way it presents itself to you, but start thinking in terms of journal entries and doing quests out of order, are you looking for a cheat.

Why do you want to get new recruits for the Blades when you do not like their ultimatum?

You are supposed to be able to recruit new Blades members even after you've been given the kill Paarthurnax quest. I know I was able to.